%------------------------------------------------------------------------------
% File     : ITP276_2 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Uniqueness 00216_013627
% 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    : 0075_VEBT_Uniqueness_00216_013627 [Des22]

% Status   : ContradictoryAxioms
% Rating   : 0.50 v8.1.0
% Syntax   : Number of formulae    : 11177 (2558 unt;1814 typ;   0 def)
%            Number of atoms       : 26838 (8143 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 19701 (2226   ~; 357   |;2256   &)
%                                         (1856 <=>;13006  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   40 (   2 avg)
%            Number of types       :   16 (  15 usr)
%            Number of type conns  : 1646 (1290   >; 356   *;   0   +;   0  <<)
%            Number of predicates  :  298 ( 295 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1504 (1504 usr;  82 con; 0-8 aty)
%            Number of variables   : 33165 (29678   !; 748   ?;33165   :)
%                                         (2739  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 15:41:39.346
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
tff(ty_t_Record_Otuple__isomorphism,type,
    tuple_isomorphism: ( $tType * $tType * $tType ) > $tType ).

tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $tType ).

tff(ty_t_Code__Numeral_Onatural,type,
    code_natural: $tType ).

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

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

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

tff(ty_t_Old__Datatype_Onode,type,
    old_node: ( $tType * $tType ) > $tType ).

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

tff(ty_t_Typerep_Otyperep,type,
    typerep: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

% Explicit typings (1789)
tff(sy_cl_Finite__Set_Ofinite,type,
    finite_finite: 
      !>[A: $tType] : $o ).

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_Ozero,type,
    zero: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Rings_Onormalization__semidom,type,
    normal8620421768224518004emidom: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oordered__comm__semiring,type,
    ordere2520102378445227354miring: 
      !>[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_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_Real__Vector__Spaces_Ometric__space,type,
    real_V7819770556892013058_space: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oreal__algebra,type,
    real_V6157519004096292374lgebra: 
      !>[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_Rings_Osemidom__divide__unit__factor,type,
    semido2269285787275462019factor: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Rings_Onormalization__semidom__multiplicative,type,
    normal6328177297339901930cative: 
      !>[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_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
    euclid5411537665997757685th_nat: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
    bNF_Ca4139267488887388095finite: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
    bNF_Ca8387033319878233205ardSuc: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
    bNF_Ca6860139660246222851ard_of: 
      !>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
    bNF_Ca8970107618336181345der_on: 
      !>[A: $tType] : ( set(A) > fun(set(product_prod(A,A)),bool) ) ).

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

tff(sy_c_BNF__Cardinal__Order__Relation_OisCardSuc,type,
    bNF_Ca6246979054910435723ardSuc: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(set(product_prod(set(A),set(A))),bool) ) ).

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

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

tff(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
    bNF_Ca7133664381575040944arCard: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

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

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

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

tff(sy_c_BNF__Def_OfstOp,type,
    bNF_fstOp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,bool)) * fun(B,fun(C,bool)) * product_prod(A,C) ) > product_prod(A,B) ) ).

tff(sy_c_BNF__Def_Opick__middlep,type,
    bNF_pick_middlep: 
      !>[B: $tType,A: $tType,C: $tType] : ( ( fun(B,fun(A,bool)) * fun(A,fun(C,bool)) * B * C ) > A ) ).

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

tff(sy_c_BNF__Def_OsndOp,type,
    bNF_sndOp: 
      !>[C: $tType,A: $tType,B: $tType] : ( ( fun(C,fun(A,bool)) * fun(A,fun(B,bool)) * product_prod(C,B) ) > product_prod(A,B) ) ).

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

tff(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
    bNF_Gr4221423524335903396lImage: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(B,A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
    bNF_Gr7122648621184425601vImage: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).

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

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

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

tff(sy_c_BNF__Wellorder__Constructions_OordIso,type,
    bNF_Wellorder_ordIso: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
    bNF_Wellorder_ordLeq: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__Constructions_OordLess,type,
    bNF_We4044943003108391690rdLess: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

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

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
    bNF_We4791949203932849705sMinim: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) * A ) > $o ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
    bNF_We1388413361240627857o_max2: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A * A ) > A ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
    bNF_We6954850376910717587_minim: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).

tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
    bNF_Wellorder_wo_suc: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).

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

tff(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
    basic_BNF_size_sum: 
      !>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * sum_sum(A,B) ) > nat ) ).

tff(sy_c_Basic__BNFs_Ofsts,type,
    basic_fsts: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > set(A) ) ).

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

tff(sy_c_Basic__BNFs_Osnds,type,
    basic_snds: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > set(B) ) ).

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

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

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
    bit_ri4277139882892585799ns_not: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
    bit_ri4674362597316999326ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
    bit_se5824344872417868541ns_and: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
    bit_se4197421643247451524op_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
    bit_se8732182000553998342ip_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
    bit_se2239418461657761734s_mask: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
    bit_se1065995026697491101ons_or: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
    bit_se4730199178511100633sh_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
    bit_se5668285175392031749et_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
    bit_se2584673776208193580ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
    bit_se2638667681897837118et_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
    bit_se5824344971392196577ns_xor: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,bool) ) ).

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

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

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

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

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

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

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

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

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

tff(sy_c_Code__Numeral_OSuc,type,
    code_Suc: fun(code_natural,code_natural) ).

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

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

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

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

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

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

tff(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
    code_nat_of_natural: fun(code_natural,nat) ).

tff(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
    code_natural_of_nat: fun(nat,code_natural) ).

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

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

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
    comple1908693960933563346ssible: 
      !>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,bool)) * fun(A,bool) ) > $o ) ).

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

tff(sy_c_Complete__Partial__Order_Occpo__class_Oiterates,type,
    comple6359979572994053840erates: 
      !>[A: $tType] : ( fun(A,A) > set(A) ) ).

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

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

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

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

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

tff(sy_c_Complex_Ocnj,type,
    cnj: complex > complex ).

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

tff(sy_c_Complex_Ocomplex_OIm,type,
    im: complex > real ).

tff(sy_c_Complex_Ocomplex_ORe,type,
    re: complex > real ).

tff(sy_c_Complex_Ocsqrt,type,
    csqrt: complex > complex ).

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

tff(sy_c_Complex_Orcis,type,
    rcis: ( real * real ) > complex ).

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

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

tff(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd,type,
    condit622319405099724424ng_bdd: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd,type,
    condit16957441358409770ng_bdd: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),bool) ) ).

tff(sy_c_Countable_Ofrom__nat,type,
    from_nat: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Countable_Onat__to__rat__surj,type,
    nat_to_rat_surj: nat > rat ).

tff(sy_c_Countable_Onth__item,type,
    nth_item: 
      !>[A: $tType] : ( nat > set(old_node(A,product_unit)) ) ).

tff(sy_c_Countable_Onth__item__rel,type,
    nth_item_rel: fun(nat,fun(nat,bool)) ).

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

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

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

tff(sy_c_Deriv_Ohas__field__derivative,type,
    has_field_derivative: 
      !>[A: $tType] : ( ( fun(A,A) * A * filter(A) ) > $o ) ).

tff(sy_c_Deriv_Ohas__vector__derivative,type,
    has_ve8173657378732805170vative: 
      !>[B: $tType] : ( ( fun(real,B) * B * filter(real) ) > $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_Equiv__Relations_Ocongruent,type,
    equiv_congruent: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).

tff(sy_c_Equiv__Relations_Ocongruent2,type,
    equiv_congruent2: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) * fun(A,fun(B,C)) ) > $o ) ).

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

tff(sy_c_Equiv__Relations_Oequivp,type,
    equiv_equivp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

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

tff(sy_c_Equiv__Relations_Oquotient,type,
    equiv_quotient: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > set(set(A)) ) ).

tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
    euclid6346220572633701492n_size: 
      !>[A: $tType] : ( A > nat ) ).

tff(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
    euclid7384307370059645450egment: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Extended__Nat_OeSuc,type,
    extended_eSuc: extended_enat > extended_enat ).

tff(sy_c_Extended__Nat_Oenat,type,
    extended_enat2: nat > extended_enat ).

tff(sy_c_Extended__Nat_Oenat_OAbs__enat,type,
    extended_Abs_enat: option(nat) > extended_enat ).

tff(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
    extended_case_enat: 
      !>[T: $tType] : ( ( fun(nat,T) * T * extended_enat ) > T ) ).

tff(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
    extend4730790105801354508finity: 
      !>[A: $tType] : A ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Finite__Set_Ofinite,type,
    finite_finite2: 
      !>[A: $tType] : fun(set(A),bool) ).

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

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

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

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

tff(sy_c_Fun_Ocomp,type,
    comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( fun(B,C) > fun(fun(A,B),fun(A,C)) ) ).

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

tff(sy_c_Fun_Oid,type,
    id: 
      !>[A: $tType] : fun(A,A) ).

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

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

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

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

tff(sy_c_Fun__Def_Opair__leq,type,
    fun_pair_leq: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).

tff(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).

tff(sy_c_GCD_OGcd__class_OGcd,type,
    gcd_Gcd: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_GCD_OGcd__class_OLcm,type,
    gcd_Lcm: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod(int,int) ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).

tff(sy_c_GCD_Obounded__quasi__semilattice,type,
    bounde8507323023520639062attice: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).

tff(sy_c_GCD_Obounded__quasi__semilattice__set,type,
    bounde6485984586167503788ce_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).

tff(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
    bounde2362111253966948842tice_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A ) > fun(set(A),A) ) ).

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

tff(sy_c_GCD_Ogcd__class_Olcm,type,
    gcd_lcm: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

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

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

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

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

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

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

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

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

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

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

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_Osemigroup,type,
    semigroup: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

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(fun(C,A),fun(set(C),A)) ).

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

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

tff(sy_c_Groups__Big_Ocomm__monoid__set,type,
    groups778175481326437816id_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
    groups_comm_monoid_F: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(fun(B,A),fun(set(B),A)) ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
    groups_comm_monoid_G: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(fun(B,A),fun(set(B),A)) ) ).

tff(sy_c_Groups__List_Ocomm__monoid__list,type,
    groups1828464146339083142d_list: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__List_Ocomm__monoid__list__set,type,
    groups4802862169904069756st_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(A,fun(list(B),A))) ).

tff(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
    groups8242544230860333062m_list: 
      !>[A: $tType] : fun(list(A),A) ).

tff(sy_c_Groups__List_Omonoid__list,type,
    groups_monoid_list: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__List_Omonoid__list_OF,type,
    groups_monoid_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(list(A),A) ) ).

tff(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
    groups5270119922927024881d_list: 
      !>[A: $tType] : fun(list(A),A) ).

tff(sy_c_HOL_ONO__MATCH,type,
    nO_MATCH: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

tff(sy_c_HOL_OThe,type,
    the: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_HOL_Oundefined,type,
    undefined: 
      !>[A: $tType] : A ).

tff(sy_c_Hilbert__Choice_Obijection,type,
    hilbert_bijection: 
      !>[A: $tType] : ( fun(A,A) > $o ) ).

tff(sy_c_Hilbert__Choice_Oinv__into,type,
    hilbert_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(B,A) ) ).

tff(sy_c_If,type,
    if: 
      !>[A: $tType] : ( ( bool * A * A ) > A ) ).

tff(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
    complete_lattice_gfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
    complete_lattice_lfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
    infini527867602293511546merate: 
      !>[A: $tType] : ( ( set(A) * nat ) > A ) ).

tff(sy_c_Int_OAbs__Integ,type,
    abs_Integ: fun(product_prod(nat,nat),int) ).

tff(sy_c_Int_ORep__Integ,type,
    rep_Integ: fun(int,product_prod(nat,nat)) ).

tff(sy_c_Int_Ointrel,type,
    intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).

tff(sy_c_Int_Onat,type,
    nat2: int > nat ).

tff(sy_c_Int_Opcr__int,type,
    pcr_int: fun(product_prod(nat,nat),fun(int,bool)) ).

tff(sy_c_Int_Opower__int,type,
    power_int: 
      !>[A: $tType] : ( ( A * int ) > A ) ).

tff(sy_c_Int_Oring__1__class_OInts,type,
    ring_1_Ints: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Int_Oring__1__class_Oof__int,type,
    ring_1_of_int: 
      !>[A: $tType] : fun(int,A) ).

tff(sy_c_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices_Osemilattice__neutr,type,
    semilattice_neutr: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices_Osemilattice__order,type,
    semilattice_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder_OMin,type,
    lattices_Min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
    lattices_ord_arg_max: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
    lattices_ord_arg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
    lattic7623131987881927897min_on: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Ois__arg__max,type,
    lattic501386751176901750rg_max: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) * B ) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
    lattic5214292709420241887eutr_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * set(A) ) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__order__set,type,
    lattic4895041142388067077er_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__set,type,
    lattic149705377957585745ce_set: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__set_OF,type,
    lattic1715443433743089157tice_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * set(A) ) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
    lattic5882676163264333800up_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Limits_OBfun,type,
    bfun: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Limits_OZfun,type,
    zfun: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Limits_Oat__infinity,type,
    at_infinity: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_List_Oappend,type,
    append: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oarg__min__list,type,
    arg_min_list: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > A ) ).

tff(sy_c_List_Oarg__min__list__rel,type,
    arg_min_list_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),list(A)),fun(product_prod(fun(A,B),list(A)),bool)) ).

tff(sy_c_List_Obutlast,type,
    butlast: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_List_Oconcat,type,
    concat: 
      !>[A: $tType] : ( list(list(A)) > list(A) ) ).

tff(sy_c_List_Ocount__list,type,
    count_list: 
      !>[A: $tType] : ( list(A) > fun(A,nat) ) ).

tff(sy_c_List_Odistinct,type,
    distinct: 
      !>[A: $tType] : ( list(A) > $o ) ).

tff(sy_c_List_Odrop,type,
    drop: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OdropWhile,type,
    dropWhile: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > option(A) ) ).

tff(sy_c_List_Ofold,type,
    fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) * B ) > B ) ).

tff(sy_c_List_Ofolding__insort__key,type,
    folding_insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * 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) * B ) > B ) ).

tff(sy_c_List_Ogen__length,type,
    gen_length: 
      !>[A: $tType] : ( nat > fun(list(A),nat) ) ).

tff(sy_c_List_Olast,type,
    last: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Olenlex,type,
    lenlex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olex,type,
    lex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexn,type,
    lexn: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * nat ) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexord,type,
    lexord: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(B,A) * set(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
    linord329482645794927042rt_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Oinsort__key,type,
    linorder_insort_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Osort__key,type,
    linorder_sort_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
    linord4507533701916653071of_set: 
      !>[A: $tType] : ( set(A) > list(A) ) ).

tff(sy_c_List_Olist_OCons,type,
    cons: 
      !>[A: $tType] : fun(A,fun(list(A),list(A))) ).

tff(sy_c_List_Olist_ONil,type,
    nil: 
      !>[A: $tType] : list(A) ).

tff(sy_c_List_Olist_Ocase__list,type,
    case_list: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) * list(A) ) > B ) ).

tff(sy_c_List_Olist_Ohd,type,
    hd: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Olist_Olist__all,type,
    list_all: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).

tff(sy_c_List_Olist_Olist__all2,type,
    list_all2: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).

tff(sy_c_List_Olist_Omap,type,
    map: 
      !>[A: $tType,Aa: $tType] : ( ( fun(A,Aa) * 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) * list(A) ) > nat ) ).

tff(sy_c_List_Olist_Otl,type,
    tl: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Olist__ex,type,
    list_ex: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).

tff(sy_c_List_Olist__update,type,
    list_update: 
      !>[A: $tType] : ( ( list(A) * nat * A ) > list(A) ) ).

tff(sy_c_List_Olistrel,type,
    listrel: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(list(A),list(B))) ) ).

tff(sy_c_List_Olistrel1,type,
    listrel1: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olistrelp,type,
    listrelp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).

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_Omeasures,type,
    measures: 
      !>[A: $tType] : ( list(fun(A,nat)) > set(product_prod(A,A)) ) ).

tff(sy_c_List_On__lists,type,
    n_lists: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).

tff(sy_c_List_Onth,type,
    nth: 
      !>[A: $tType] : ( list(A) > fun(nat,A) ) ).

tff(sy_c_List_Onths,type,
    nths: 
      !>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).

tff(sy_c_List_Oproduct,type,
    product: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_List_Oproduct__lists,type,
    product_lists: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Oremdups,type,
    remdups: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oremdups__adj,type,
    remdups_adj: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oremove1,type,
    remove1: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

tff(sy_c_List_OremoveAll,type,
    removeAll: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

tff(sy_c_List_Oreplicate,type,
    replicate: 
      !>[A: $tType] : ( ( nat * A ) > list(A) ) ).

tff(sy_c_List_Orev,type,
    rev: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Orotate,type,
    rotate: 
      !>[A: $tType] : ( nat > fun(list(A),list(A)) ) ).

tff(sy_c_List_Orotate1,type,
    rotate1: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oshuffles,type,
    shuffles: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > set(list(A)) ) ).

tff(sy_c_List_Osorted__wrt,type,
    sorted_wrt: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) ) > $o ) ).

tff(sy_c_List_Osplice,type,
    splice: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Osubseqs,type,
    subseqs: 
      !>[A: $tType] : ( list(A) > list(list(A)) ) ).

tff(sy_c_List_Otake,type,
    take: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OtakeWhile,type,
    takeWhile: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(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_Ozip,type,
    zip: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_Map_Odom,type,
    dom: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(A) ) ).

tff(sy_c_Map_Ograph,type,
    graph: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).

tff(sy_c_Map_Omap__add,type,
    map_add: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__comp,type,
    map_comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,option(C)) * fun(A,option(B)) * A ) > option(C) ) ).

tff(sy_c_Map_Omap__le,type,
    map_le: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > $o ) ).

tff(sy_c_Map_Omap__of,type,
    map_of: 
      !>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__upds,type,
    map_upds: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) * list(B) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

tff(sy_c_Map_Orestrict__map,type,
    restrict_map: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

tff(sy_c_Nat_Ocompow,type,
    compow: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Nat_Ofunpow,type,
    funpow: 
      !>[A: $tType] : fun(nat,fun(fun(A,A),fun(A,A))) ).

tff(sy_c_Nat_Onat_Ocase__nat,type,
    case_nat: 
      !>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
    rec_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).

tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
    rec_set_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,bool) ) ).

tff(sy_c_Nat_Osemiring__1__class_ONats,type,
    semiring_1_Nats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
    semiring_1_of_nat: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
    semiri8178284476397505188at_aux: 
      !>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).

tff(sy_c_Nat_Osize__class_Osize,type,
    size_size: 
      !>[A: $tType] : fun(A,nat) ).

tff(sy_c_Nat__Bijection_Oint__decode,type,
    nat_int_decode: fun(nat,int) ).

tff(sy_c_Nat__Bijection_Oint__encode,type,
    nat_int_encode: fun(int,nat) ).

tff(sy_c_Nat__Bijection_Olist__decode,type,
    nat_list_decode: fun(nat,list(nat)) ).

tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: fun(list(nat),nat) ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun(list(nat),fun(list(nat),bool)) ).

tff(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: fun(nat,product_prod(nat,nat)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > fun(nat,product_prod(nat,nat)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun(product_prod(nat,nat),nat) ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun(set(nat),nat) ).

tff(sy_c_Nat__Bijection_Osum__decode,type,
    nat_sum_decode: fun(nat,sum_sum(nat,nat)) ).

tff(sy_c_Nat__Bijection_Osum__encode,type,
    nat_sum_encode: fun(sum_sum(nat,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: real > real ).

tff(sy_c_Num_OBitM,type,
    bitM: num > num ).

tff(sy_c_Num_Oinc,type,
    inc: num > num ).

tff(sy_c_Num_Onat__of__num,type,
    nat_of_num: fun(num,nat) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl,type,
    neg_numeral_dbl: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
    neg_numeral_dbl_dec: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
    neg_numeral_dbl_inc: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Ois__num,type,
    neg_numeral_is_num: 
      !>[A: $tType] : ( A > $o ) ).

tff(sy_c_Num_Oneg__numeral__class_Osub,type,
    neg_numeral_sub: 
      !>[A: $tType] : ( ( num * num ) > A ) ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_Ocase__num,type,
    case_num: 
      !>[A: $tType] : fun(A,fun(fun(num,A),fun(fun(num,A),fun(num,A)))) ).

tff(sy_c_Num_Onum_Orec__num,type,
    rec_num: 
      !>[A: $tType] : fun(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)))) ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : fun(num,A) ).

tff(sy_c_Num_Opow,type,
    pow: ( num * num ) > num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

tff(sy_c_Num_Oring__1__class_Oiszero,type,
    ring_1_iszero: 
      !>[A: $tType] : ( A > $o ) ).

tff(sy_c_Num_Osqr,type,
    sqr: num > num ).

tff(sy_c_Old__Datatype_OAtom,type,
    old_Atom: 
      !>[A: $tType,B: $tType] : ( sum_sum(A,nat) > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OIn0,type,
    old_In0: 
      !>[A: $tType,B: $tType] : ( set(old_node(A,B)) > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OIn1,type,
    old_In1: 
      !>[A: $tType,B: $tType] : ( set(old_node(A,B)) > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OLeaf,type,
    old_Leaf: 
      !>[A: $tType,B: $tType] : ( A > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_ONode,type,
    old_Node: 
      !>[B: $tType,A: $tType] : set(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))) ).

tff(sy_c_Old__Datatype_ONumb,type,
    old_Numb: 
      !>[A: $tType,B: $tType] : ( nat > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OPush,type,
    old_Push: 
      !>[B: $tType] : ( ( sum_sum(B,nat) * fun(nat,sum_sum(B,nat)) * nat ) > sum_sum(B,nat) ) ).

tff(sy_c_Old__Datatype_OPush__Node,type,
    old_Push_Node: 
      !>[B: $tType,A: $tType] : ( sum_sum(B,nat) > fun(old_node(A,B),old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OScons,type,
    old_Scons: 
      !>[A: $tType,B: $tType] : ( ( set(old_node(A,B)) * set(old_node(A,B)) ) > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_Ondepth,type,
    old_ndepth: 
      !>[A: $tType,B: $tType] : ( old_node(A,B) > nat ) ).

tff(sy_c_Old__Datatype_Onode_OAbs__Node,type,
    old_Abs_Node: 
      !>[B: $tType,A: $tType] : ( product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)) > old_node(A,B) ) ).

tff(sy_c_Old__Datatype_Onode_ORep__Node,type,
    old_Rep_Node: 
      !>[A: $tType,B: $tType] : ( old_node(A,B) > product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)) ) ).

tff(sy_c_Old__Datatype_Ontrunc,type,
    old_ntrunc: 
      !>[A: $tType,B: $tType] : ( ( nat * set(old_node(A,B)) ) > set(old_node(A,B)) ) ).

tff(sy_c_Option_Obind,type,
    bind: 
      !>[A: $tType,B: $tType] : fun(option(A),fun(fun(A,option(B)),option(B))) ).

tff(sy_c_Option_Ocombine__options,type,
    combine_options: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * option(A) * option(A) ) > option(A) ) ).

tff(sy_c_Option_Ois__none,type,
    is_none: 
      !>[A: $tType] : ( option(A) > $o ) ).

tff(sy_c_Option_Ooption_ONone,type,
    none: 
      !>[A: $tType] : option(A) ).

tff(sy_c_Option_Ooption_OSome,type,
    some: 
      !>[A: $tType] : fun(A,option(A)) ).

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : fun(B,fun(fun(A,B),fun(option(A),B))) ).

tff(sy_c_Option_Ooption_Omap__option,type,
    map_option: 
      !>[A: $tType,Aa: $tType] : fun(fun(A,Aa),fun(option(A),option(Aa))) ).

tff(sy_c_Option_Ooption_Opred__option,type,
    pred_option: 
      !>[A: $tType] : fun(fun(A,bool),fun(option(A),bool)) ).

tff(sy_c_Option_Ooption_Orec__option,type,
    rec_option: 
      !>[C: $tType,A: $tType] : fun(C,fun(fun(A,C),fun(option(A),C))) ).

tff(sy_c_Option_Ooption_Orel__option,type,
    rel_option: 
      !>[A: $tType,B: $tType] : fun(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool))) ).

tff(sy_c_Option_Ooption_Oset__option,type,
    set_option: 
      !>[A: $tType] : fun(option(A),set(A)) ).

tff(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( fun(A,nat) > fun(option(A),nat) ) ).

tff(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : fun(option(A),A) ).

tff(sy_c_Option_Othese,type,
    these: 
      !>[A: $tType] : ( set(option(A)) > set(A) ) ).

tff(sy_c_Order__Continuity_Ocountable__complete__lattice__class_Occlfp,type,
    order_532582986084564980_cclfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Order__Continuity_Oinf__continuous,type,
    order_inf_continuous: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Order__Continuity_Osup__continuous,type,
    order_sup_continuous: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Order__Relation_OAbove,type,
    order_Above: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_OAboveS,type,
    order_AboveS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_OUnder,type,
    order_Under: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_OUnderS,type,
    order_UnderS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_Oabove,type,
    order_above: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_OaboveS,type,
    order_aboveS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_Olinear__order__on,type,
    order_679001287576687338der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Oofilter,type,
    order_ofilter: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > $o ) ).

tff(sy_c_Order__Relation_Opartial__order__on,type,
    order_7125193373082350890der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Opreorder__on,type,
    order_preorder_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Orelation__of,type,
    order_relation_of: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Order__Relation_Ounder,type,
    order_under: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_OunderS,type,
    order_underS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_Owell__order__on,type,
    order_well_order_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Orderings_Obot__class_Obot,type,
    bot_bot: 
      !>[A: $tType] : A ).

tff(sy_c_Orderings_Oord_OLeast,type,
    least: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(fun(A,bool),A) ) ).

tff(sy_c_Orderings_Oord_Omax,type,
    max: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_Orderings_Oord__class_Omax,type,
    ord_max: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_Orderings_Oorder__class_Oantimono,type,
    order_antimono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Omono,type,
    order_mono: 
      !>[A: $tType,B: $tType] : fun(fun(A,B),bool) ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
    order_strict_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oordering,type,
    ordering: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Orderings_Oordering__axioms,type,
    ordering_axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Orderings_Oordering__top,type,
    ordering_top: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * A ) > $o ) ).

tff(sy_c_Orderings_Oordering__top__axioms,type,
    ordering_top_axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > $o ) ).

tff(sy_c_Orderings_Opartial__preordering,type,
    partial_preordering: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Orderings_Opreordering,type,
    preordering: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Orderings_Opreordering__axioms,type,
    preordering_axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

tff(sy_c_Power_Opower_Opower,type,
    power2: 
      !>[A: $tType] : ( ( A * fun(A,fun(A,A)) * A * nat ) > A ) ).

tff(sy_c_Power_Opower__class_Opower,type,
    power_power: 
      !>[A: $tType] : fun(A,fun(nat,A)) ).

tff(sy_c_Product__Type_OPair,type,
    product_Pair: 
      !>[A: $tType,B: $tType] : fun(A,fun(B,product_prod(A,B))) ).

tff(sy_c_Product__Type_OSigma,type,
    product_Sigma: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Product__Type_Oapfst,type,
    product_apfst: 
      !>[A: $tType,C: $tType,B: $tType] : ( fun(A,C) > fun(product_prod(A,B),product_prod(C,B)) ) ).

tff(sy_c_Product__Type_Oapsnd,type,
    product_apsnd: 
      !>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(product_prod(A,B),product_prod(A,C))) ).

tff(sy_c_Product__Type_Ocurry,type,
    product_curry: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(product_prod(A,B),C) > fun(A,fun(B,C)) ) ).

tff(sy_c_Product__Type_Ointernal__case__prod,type,
    produc5280177257484947105e_prod: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).

tff(sy_c_Product__Type_Omap__prod,type,
    product_map_prod: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(product_prod(A,B),product_prod(C,D)) ) ).

tff(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
    product_rec_bool: 
      !>[T: $tType] : ( ( T * T * bool ) > T ) ).

tff(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
    product_rec_set_bool: 
      !>[T: $tType] : ( ( T * T * bool ) > fun(T,bool) ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
    product_rec_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
    product_rec_set_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > fun(T,bool) ) ).

tff(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
    product_rec_set_unit: 
      !>[T: $tType] : ( ( T * product_unit ) > fun(T,bool) ) ).

tff(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
    product_rec_unit: 
      !>[T: $tType] : ( ( T * product_unit ) > T ) ).

tff(sy_c_Product__Type_Oprod_Ocase__prod,type,
    product_case_prod: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).

tff(sy_c_Product__Type_Oprod_Ofst,type,
    product_fst: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).

tff(sy_c_Product__Type_Oprod_Osnd,type,
    product_snd: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).

tff(sy_c_Product__Type_Oprod_Oswap,type,
    product_swap: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),product_prod(B,A)) ).

tff(sy_c_Product__Type_Oproduct,type,
    product_product: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Product__Type_Oscomp,type,
    product_scomp: 
      !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,product_prod(B,C)) * fun(B,fun(C,D)) ) > fun(A,D) ) ).

tff(sy_c_Pure_Otype,type,
    type2: 
      !>[A: $tType] : itself(A) ).

tff(sy_c_Random_Oiterate,type,
    iterate: 
      !>[B: $tType,A: $tType] : ( ( code_natural * fun(B,fun(A,product_prod(B,A))) ) > fun(B,fun(A,product_prod(B,A))) ) ).

tff(sy_c_Random_Oiterate__rel,type,
    iterate_rel: 
      !>[B: $tType,A: $tType] : fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool)) ).

tff(sy_c_Random_Olog,type,
    log: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Ominus__shift,type,
    minus_shift: ( code_natural * code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Onext,type,
    next: fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).

tff(sy_c_Random_Opick,type,
    pick: 
      !>[A: $tType] : ( ( list(product_prod(code_natural,A)) * code_natural ) > A ) ).

tff(sy_c_Random_Orange,type,
    range: code_natural > fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).

tff(sy_c_Random_Oselect,type,
    select: 
      !>[A: $tType] : ( list(A) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: fun(product_prod(int,int),rat) ).

tff(sy_c_Rat_OFract,type,
    fract: ( int * int ) > rat ).

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun(rat,product_prod(int,int)) ).

tff(sy_c_Rat_Ofield__char__0__class_ORats,type,
    field_char_0_Rats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
    field_char_0_of_rat: 
      !>[A: $tType] : ( rat > A ) ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun(product_prod(int,int),fun(rat,bool)) ).

tff(sy_c_Rat_Opositive,type,
    positive: fun(rat,bool) ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod(int,int) ).

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

tff(sy_c_Real_ORatreal,type,
    ratreal: rat > real ).

tff(sy_c_Real_OReal,type,
    real2: fun(nat,rat) > real ).

tff(sy_c_Real_Ocauchy,type,
    cauchy: fun(nat,rat) > $o ).

tff(sy_c_Real_Opcr__real,type,
    pcr_real: fun(fun(nat,rat),fun(real,bool)) ).

tff(sy_c_Real_Opositive,type,
    positive2: fun(real,bool) ).

tff(sy_c_Real_Orealrel,type,
    realrel: fun(fun(nat,rat),fun(fun(nat,rat),bool)) ).

tff(sy_c_Real_Orep__real,type,
    rep_real: fun(real,fun(nat,rat)) ).

tff(sy_c_Real_Ovanishes,type,
    vanishes: fun(nat,rat) > $o ).

tff(sy_c_Real__Vector__Spaces_OReals,type,
    real_Vector_Reals: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Real__Vector__Spaces_Obounded__bilinear,type,
    real_V2442710119149674383linear: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear,type,
    real_V3181309239436604168linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear__axioms,type,
    real_V4916620083959148203axioms: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Oconstruct,type,
    real_V4425403222259421789struct: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * A ) > B ) ).

tff(sy_c_Real__Vector__Spaces_Odependent,type,
    real_V358717886546972837endent: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Odim,type,
    real_Vector_dim: 
      !>[A: $tType] : ( set(A) > nat ) ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
    real_V557655796197034286t_dist: 
      !>[A: $tType] : ( ( A * A ) > real ) ).

tff(sy_c_Real__Vector__Spaces_Oextend__basis,type,
    real_V4986007116245087402_basis: 
      !>[A: $tType] : ( set(A) > set(A) ) ).

tff(sy_c_Real__Vector__Spaces_Olinear,type,
    real_Vector_linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
    real_V7770717601297561774m_norm: 
      !>[A: $tType] : ( A > real ) ).

tff(sy_c_Real__Vector__Spaces_Oof__real,type,
    real_Vector_of_real: 
      !>[A: $tType] : ( real > A ) ).

tff(sy_c_Real__Vector__Spaces_Orepresentation,type,
    real_V7696804695334737415tation: 
      !>[A: $tType] : ( ( set(A) * A * A ) > real ) ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
    real_V8093663219630862766scaleR: 
      !>[A: $tType] : fun(real,fun(A,A)) ).

tff(sy_c_Real__Vector__Spaces_Ospan,type,
    real_Vector_span: 
      !>[A: $tType] : ( set(A) > set(A) ) ).

tff(sy_c_Real__Vector__Spaces_Osubspace,type,
    real_Vector_subspace: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Record_Otuple__isomorphism_OTuple__Isomorphism,type,
    tuple_1188178415141063261rphism: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,product_prod(B,C)) * fun(product_prod(B,C),A) ) > tuple_isomorphism(A,B,C) ) ).

tff(sy_c_Record_Otuple__isomorphism_Osize__tuple__isomorphism,type,
    tuple_9181185373184732606rphism: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,nat) * fun(B,nat) * fun(C,nat) * tuple_isomorphism(A,B,C) ) > nat ) ).

tff(sy_c_Relation_ODomain,type,
    domain: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(A) ) ).

tff(sy_c_Relation_OField,type,
    field2: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).

tff(sy_c_Relation_OId,type,
    id2: 
      !>[A: $tType] : set(product_prod(A,A)) ).

tff(sy_c_Relation_OId__on,type,
    id_on: 
      !>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).

tff(sy_c_Relation_OImage,type,
    image: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,B)) * set(A) ) > set(B) ) ).

tff(sy_c_Relation_ORange,type,
    range2: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(B) ) ).

tff(sy_c_Relation_ORangep,type,
    rangep: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,bool) ) ).

tff(sy_c_Relation_Oantisym,type,
    antisym: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Oantisymp,type,
    antisymp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Relation_Oconverse,type,
    converse: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(B,A)) ) ).

tff(sy_c_Relation_Oconversep,type,
    conversep: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,fun(A,bool)) ) ).

tff(sy_c_Relation_Oinv__image,type,
    inv_image: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Relation_Oirrefl,type,
    irrefl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Oirreflp,type,
    irreflp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Relation_Orefl__on,type,
    refl_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Relation_Oreflp,type,
    reflp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Relation_Orelcomp,type,
    relcomp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).

tff(sy_c_Relation_Orelcompp,type,
    relcompp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,bool)) * fun(B,fun(C,bool)) ) > fun(A,fun(C,bool)) ) ).

tff(sy_c_Relation_Osingle__valued,type,
    single_valued: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).

tff(sy_c_Relation_Osingle__valuedp,type,
    single_valuedp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Relation_Ototal__on,type,
    total_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Relation_Otrans,type,
    trans: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Otransp,type,
    transp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
    algebr8660921524188924756oprime: 
      !>[A: $tType] : ( ( A * A ) > $o ) ).

tff(sy_c_Rings_Odivide__class_Odivide,type,
    divide_divide: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
    normal6383669964737779283malize: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
    unit_f5069060285200089521factor: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun(bool,A) ).

tff(sy_c_Series_Osuminf,type,
    suminf: 
      !>[A: $tType] : ( fun(nat,A) > A ) ).

tff(sy_c_Series_Osummable,type,
    summable: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Series_Osums,type,
    sums: 
      !>[A: $tType] : ( ( fun(nat,A) * A ) > $o ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : fun(fun(A,bool),set(A)) ).

tff(sy_c_Set_OPow,type,
    pow2: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Odisjnt,type,
    disjnt: 
      !>[A: $tType] : ( ( set(A) * set(A) ) > $o ) ).

tff(sy_c_Set_Oimage,type,
    image2: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > set(B) ) ).

tff(sy_c_Set_Oinsert,type,
    insert: 
      !>[A: $tType] : ( A > fun(set(A),set(A)) ) ).

tff(sy_c_Set_Ovimage,type,
    vimage: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(B) ) > set(A) ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
    set_fo1817059534552279752at_rel: 
      !>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool)) ).

tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

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_Sum__Type_OInl,type,
    sum_Inl: 
      !>[A: $tType,B: $tType] : fun(A,sum_sum(A,B)) ).

tff(sy_c_Sum__Type_OInr,type,
    sum_Inr: 
      !>[B: $tType,A: $tType] : fun(B,sum_sum(A,B)) ).

tff(sy_c_Sum__Type_OPlus,type,
    sum_Plus: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(sum_sum(A,B)) ) ).

tff(sy_c_Sum__Type_Osum_Ocase__sum,type,
    sum_case_sum: 
      !>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(B,C) ) > fun(sum_sum(A,B),C) ) ).

tff(sy_c_Topological__Spaces_Ocontinuous,type,
    topolo3448309680560233919inuous: 
      !>[A: $tType,B: $tType] : ( ( filter(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Topological__Spaces_Ocontinuous__on,type,
    topolo81223032696312382ous_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Topological__Spaces_Omonoseq,type,
    topological_monoseq: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
    topolo1002775350975398744n_open: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
    topolo3827282254853284352ce_Lim: 
      !>[F: $tType,A: $tType] : ( ( filter(F) * fun(F,A) ) > A ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
    topolo174197925503356063within: 
      !>[A: $tType] : ( ( A * set(A) ) > filter(A) ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oclosed,type,
    topolo7761053866217962861closed: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Ocompact,type,
    topolo2193935891317330818ompact: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oconnected,type,
    topolo1966860045006549960nected: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent,type,
    topolo6863149650580417670ergent: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
    topolo7230453075368039082e_nhds: 
      !>[A: $tType] : ( A > filter(A) ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
    topolo3814608138187158403Cauchy: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
    topolo6688025880775521714ounded: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity,type,
    topolo7806501430040627800ormity: 
      !>[A: $tType] : filter(product_prod(A,A)) ).

tff(sy_c_Topological__Spaces_Ouniformly__continuous__on,type,
    topolo6026614971017936543ous_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: real > real ).

tff(sy_c_Transcendental_Oarcosh,type,
    arcosh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: real > real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: real > real ).

tff(sy_c_Transcendental_Oarsinh,type,
    arsinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oartanh,type,
    artanh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocos,type,
    cos: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: nat > real ).

tff(sy_c_Transcendental_Ocosh,type,
    cosh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocot,type,
    cot: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Odiffs,type,
    diffs: 
      !>[A: $tType] : ( fun(nat,A) > fun(nat,A) ) ).

tff(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oln__class_Oln,type,
    ln_ln: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Olog,type,
    log2: real > fun(real,real) ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr,type,
    powr: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Transcendental_Osin,type,
    sin: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transfer_Obi__total,type,
    bi_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Transitive__Closure_Oacyclic,type,
    transitive_acyclic: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Transitive__Closure_Ontrancl,type,
    transitive_ntrancl: 
      !>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Ortrancl,type,
    transitive_rtrancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Ortranclp,type,
    transitive_rtranclp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,bool)) ) ).

tff(sy_c_Transitive__Closure_Otrancl,type,
    transitive_trancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Otranclp,type,
    transitive_tranclp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,bool)) ) ).

tff(sy_c_Typerep_Otyperep_OTyperep,type,
    typerep2: ( literal * list(typerep) ) > typerep ).

tff(sy_c_Typerep_Otyperep_Osize__typerep,type,
    size_typerep: fun(typerep,nat) ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( bool * bool ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option(product_prod(nat,nat)) * nat * list(vEBT_VEBT) * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Ocase__VEBT,type,
    vEBT_case_VEBT: 
      !>[A: $tType] : fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A))) ).

tff(sy_c_VEBT__Definitions_OVEBT_Orec__VEBT,type,
    vEBT_rec_VEBT: 
      !>[A: $tType] : fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A))) ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun(vEBT_VEBT,fun(nat,bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
    vEBT_VEBT_elim_dead: ( vEBT_VEBT * extended_enat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: fun(product_prod(vEBT_VEBT,extended_enat),fun(product_prod(vEBT_VEBT,extended_enat),bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list(vEBT_VEBT) * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__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),bool)) ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat * nat ) > nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > bool ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun(nat,bool) ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: ( option(nat) * option(nat) ) > bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: ( option(nat) * option(nat) ) > bool ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__comp__shift,type,
    vEBT_V6923181176774028177_shift: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * option(A) * option(A) ) > $o ) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__comp__shift__rel,type,
    vEBT_V4810408830578336424ft_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),fun(product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),bool)) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
    vEBT_V2048590022279873568_shift: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(option(A),fun(option(A),option(A))) ) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel,type,
    vEBT_V459564278314245337ft_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),fun(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),bool)) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,bool)) ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),bool)) ).

tff(sy_c_Vector__Spaces_Olinear,type,
    vector_linear: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,B)) * fun(A,fun(C,C)) * fun(B,C) ) > $o ) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > $o ) ).

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set(product_prod(nat,nat)) ).

tff(sy_c_Wellfounded_Olex__prod,type,
    lex_prod: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_Wellfounded_Omax__ext,type,
    max_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omax__extp,type,
    max_extp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),fun(set(A),bool)) ) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Omin__ext,type,
    min_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set(product_prod(nat,nat)) ).

tff(sy_c_Wellfounded_Owf,type,
    wf: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Wellfounded_OwfP,type,
    wfP: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Wfrec_Ocut,type,
    cut: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(product_prod(A,A)) * A ) > fun(A,B) ) ).

tff(sy_c_Wfrec_Osame__fst,type,
    same_fst: 
      !>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_Zorn_OChains,type,
    chains: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(set(A)) ) ).

tff(sy_c_Zorn_Oinit__seg__of,type,
    init_seg_of: 
      !>[A: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))) ).

tff(sy_c_Zorn_Opred__on_Ochain,type,
    pred_chain: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) ) > fun(set(A),bool) ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fAll,type,
    fAll: 
      !>[A: $tType] : ( fun(A,bool) > bool ) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_fEx,type,
    fEx: 
      !>[A: $tType] : fun(fun(A,bool),bool) ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun(bool,bool) ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: ( bool * bool ) > bool ).

tff(sy_c_fdisj,type,
    fdisj: ( bool * bool ) > bool ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_fimplies,type,
    fimplies: fun(bool,fun(bool,bool)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : fun(A,fun(set(A),bool)) ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_a____,type,
    a: bool ).

tff(sy_v_b____,type,
    b: bool ).

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_info____,type,
    info: option(product_prod(nat,nat)) ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_sa____,type,
    sa: vEBT_VEBT ).

tff(sy_v_summary_H____,type,
    summary: vEBT_VEBT ).

tff(sy_v_summary____,type,
    summary2: vEBT_VEBT ).

tff(sy_v_treeList_H____,type,
    treeList: list(vEBT_VEBT) ).

tff(sy_v_treeList____,type,
    treeList2: list(vEBT_VEBT) ).

% Relevant facts (8477)
tff(fact_0_case4_I9_J,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),mi),ma)) ).

% case4(9)
tff(fact_1_option_Oinject,axiom,
    ! [A: $tType,X2: A,Y2: A] :
      ( ( aa(A,option(A),some(A),X2) = aa(A,option(A),some(A),Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_2_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_3_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
    <=> ( ( A3 = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
tff(fact_4_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod(nat,nat)] :
      ~ ! [K: nat,M: nat] : X != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K),M) ).

% prod_decode_aux.cases
tff(fact_5_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y: product_prod(A,B)] :
      ~ ! [A5: A,B4: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) ).

% old.prod.exhaust
tff(fact_6_surj__pair,axiom,
    ! [A: $tType,B: $tType,P: product_prod(A,B)] :
    ? [X3: A,Y3: B] : P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) ).

% surj_pair
tff(fact_7_prod__cases,axiom,
    ! [B: $tType,A: $tType,P2: fun(product_prod(A,B),bool),P: product_prod(A,B)] :
      ( ! [A5: A,B4: B] : pp(aa(product_prod(A,B),bool,P2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)))
     => pp(aa(product_prod(A,B),bool,P2,P)) ) ).

% prod_cases
tff(fact_8_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
     => ~ ( ( A3 = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
tff(fact_9_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
      ~ ! [A5: A,B4: B,C2: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C2)) ).

% prod_cases3
tff(fact_10_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ~ ! [A5: A,B4: B,C2: C,D2: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A5),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D2))) ).

% prod_cases4
tff(fact_11_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A5),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))) ).

% prod_cases5
tff(fact_12_max__in__set__def,axiom,
    ! [Xs: set(nat),X: nat] :
      ( vEBT_VEBT_max_in_set(Xs,X)
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X),Xs))
        & ! [X4: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),Xs))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),X)) ) ) ) ).

% max_in_set_def
tff(fact_13_min__in__set__def,axiom,
    ! [Xs: set(nat),X: nat] :
      ( vEBT_VEBT_min_in_set(Xs,X)
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X),Xs))
        & ! [X4: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),Xs))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),X4)) ) ) ) ).

% min_in_set_def
tff(fact_14_lesseq__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y))
    <=> vEBT_VEBT_lesseq(aa(nat,option(nat),some(nat),X),aa(nat,option(nat),some(nat),Y)) ) ).

% lesseq_shift
tff(fact_15_prod__induct7,axiom,
    ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))] :
      ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),bool,P2,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),B4),aa(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F,G)))),C2),aa(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G))),aa(D,fun(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G)))),product_Pair(D,product_prod(E,product_prod(F,G))),D2),aa(product_prod(F,G),product_prod(E,product_prod(F,G)),aa(E,fun(product_prod(F,G),product_prod(E,product_prod(F,G))),product_Pair(E,product_prod(F,G)),E2),aa(G,product_prod(F,G),aa(F,fun(G,product_prod(F,G)),product_Pair(F,G),F2),G2))))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),bool,P2,X)) ) ).

% prod_induct7
tff(fact_16_prod__induct6,axiom,
    ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))] :
      ( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F2: F] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),bool,P2,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F)))),B4),aa(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F))),aa(C,fun(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F)))),product_Pair(C,product_prod(D,product_prod(E,F))),C2),aa(product_prod(E,F),product_prod(D,product_prod(E,F)),aa(D,fun(product_prod(E,F),product_prod(D,product_prod(E,F))),product_Pair(D,product_prod(E,F)),D2),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F2)))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),bool,P2,X)) ) ).

% prod_induct6
tff(fact_17_prod__induct5,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P2,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A5),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P2,X)) ) ).

% prod_induct5
tff(fact_18_prod__induct4,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool),X: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ( ! [A5: A,B4: B,C2: C,D2: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P2,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A5),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D2)))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P2,X)) ) ).

% prod_induct4
tff(fact_19_prod__induct3,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,C)),bool),X: product_prod(A,product_prod(B,C))] :
      ( ! [A5: A,B4: B,C2: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,P2,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C2))))
     => pp(aa(product_prod(A,product_prod(B,C)),bool,P2,X)) ) ).

% prod_induct3
tff(fact_20_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))] :
      ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),B4),aa(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F,G)))),C2),aa(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G))),aa(D,fun(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G)))),product_Pair(D,product_prod(E,product_prod(F,G))),D2),aa(product_prod(F,G),product_prod(E,product_prod(F,G)),aa(E,fun(product_prod(F,G),product_prod(E,product_prod(F,G))),product_Pair(E,product_prod(F,G)),E2),aa(G,product_prod(F,G),aa(F,fun(G,product_prod(F,G)),product_Pair(F,G),F2),G2)))))) ).

% prod_cases7
tff(fact_21_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))] :
      ~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F2: F] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F)))),B4),aa(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F))),aa(C,fun(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F)))),product_Pair(C,product_prod(D,product_prod(E,F))),C2),aa(product_prod(E,F),product_prod(D,product_prod(E,F)),aa(D,fun(product_prod(E,F),product_prod(D,product_prod(E,F))),product_Pair(D,product_prod(E,F)),D2),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F2))))) ).

% prod_cases6
tff(fact_22_old_Oprod_Orec,axiom,
    ! [A: $tType,T: $tType,B: $tType,F1: fun(A,fun(B,T)),A3: A,B2: B] : product_rec_prod(A,B,T,F1,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) = aa(B,T,aa(A,fun(B,T),F1,A3),B2) ).

% old.prod.rec
tff(fact_23_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X)) ) ).

% order_refl
tff(fact_24_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),A3)) ) ).

% dual_order.refl
tff(fact_25_internal__case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),produc5280177257484947105e_prod(B,C,A),C3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),C3,A3),B2) ).

% internal_case_prod_conv
tff(fact_26_relChain__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [R: set(product_prod(A,A)),As: fun(A,B)] :
          ( bNF_Ca3754400796208372196lChain(A,B,R,As)
        <=> ! [I: A,J: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J)),R))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,As,I)),aa(A,B,As,J))) ) ) ) ).

% relChain_def
tff(fact_27_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F2: fun(nat,fun(A,A)),A5: nat,B4: nat,Acc: A] : X != aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),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)),A5),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B4),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_28_le__prod__encode__1,axiom,
    ! [A3: nat,B2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),B2)))) ).

% le_prod_encode_1
tff(fact_29_le__prod__encode__2,axiom,
    ! [B2: nat,A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),B2)))) ).

% le_prod_encode_2
tff(fact_30_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N: nat,N2: nat] :
          ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N3)),aa(nat,A,F3,aa(nat,nat,suc,N3))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N)),aa(nat,A,F3,N2))) ) ) ) ).

% lift_Suc_mono_le
tff(fact_31_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N: nat,N2: nat] :
          ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N3))),aa(nat,A,F3,N3)))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N2)),aa(nat,A,F3,N))) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_32_le__refl,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N)) ).

% le_refl
tff(fact_33_le__trans,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),K2)) ) ) ).

% le_trans
tff(fact_34__C1_C_I2_J,axiom,
    deg = aa(nat,nat,suc,zero_zero(nat)) ).

% "1"(2)
tff(fact_35_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_36_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X2) = aa(nat,nat,suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_37_prod__encode__eq,axiom,
    ! [X: product_prod(nat,nat),Y: product_prod(nat,nat)] :
      ( ( aa(product_prod(nat,nat),nat,nat_prod_encode,X) = aa(product_prod(nat,nat),nat,nat_prod_encode,Y) )
    <=> ( X = Y ) ) ).

% prod_encode_eq
tff(fact_38_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(nat,nat,suc,M2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2)) ) ).

% Suc_le_mono
tff(fact_39_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa(nat,nat,suc,N) ).

% n_not_Suc_n
tff(fact_40_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_41_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R2: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( ! [X3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X3),X3))
       => ( ! [X3: nat,Y3: nat,Z: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X3),Y3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,Y3),Z))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X3),Z)) ) )
         => ( ! [N3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,N3),aa(nat,nat,suc,N3)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,M2),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_42_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( pp(aa(nat,bool,P2,M2))
       => ( ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N3))
             => ( pp(aa(nat,bool,P2,N3))
               => pp(aa(nat,bool,P2,aa(nat,nat,suc,N3))) ) )
         => pp(aa(nat,bool,P2,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_43_mem__Collect__eq,axiom,
    ! [A: $tType,A3: A,P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),aa(fun(A,bool),set(A),collect(A),P2)))
    <=> pp(aa(A,bool,P2,A3)) ) ).

% mem_Collect_eq
tff(fact_44_Collect__mem__eq,axiom,
    ! [A: $tType,A6: set(A)] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_a(set(A),fun(A,bool),A6)) = A6 ).

% Collect_mem_eq
tff(fact_45_Collect__cong,axiom,
    ! [A: $tType,P2: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P2,X3))
        <=> pp(aa(A,bool,Q,X3)) )
     => ( aa(fun(A,bool),set(A),collect(A),P2) = aa(fun(A,bool),set(A),collect(A),Q) ) ) ).

% Collect_cong
tff(fact_46_ext,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),G3: fun(A,B)] :
      ( ! [X3: A] : aa(A,B,F3,X3) = aa(A,B,G3,X3)
     => ( F3 = G3 ) ) ).

% ext
tff(fact_47_full__nat__induct,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M3)),N3))
             => pp(aa(nat,bool,P2,M3)) )
         => pp(aa(nat,bool,P2,N3)) )
     => pp(aa(nat,bool,P2,N)) ) ).

% full_nat_induct
tff(fact_48_not__less__eq__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M2)) ) ).

% not_less_eq_eq
tff(fact_49_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_50_le__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
        | ( M2 = aa(nat,nat,suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_51_Suc__le__D,axiom,
    ! [N: nat,M4: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M4))
     => ? [M: nat] : M4 = aa(nat,nat,suc,M) ) ).

% Suc_le_D
tff(fact_52_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N))) ) ).

% le_SucI
tff(fact_53_le__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( M2 = aa(nat,nat,suc,N) ) ) ) ).

% le_SucE
tff(fact_54_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% Suc_leD
tff(fact_55_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_56_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_le_cases
tff(fact_57_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B2: A,F3: fun(A,B),C3: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( ( aa(A,B,F3,B2) = C3 )
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,A3)),C3)) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_58_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C3: B] :
          ( ( A3 = aa(B,A,F3,B2) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C3))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F3,C3))) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_59_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_linear
tff(fact_60_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% order_eq_refl
tff(fact_61_order__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F3: fun(A,C),C3: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,B2)),C3))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,X3)),aa(A,C,F3,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,A3)),C3)) ) ) ) ) ).

% order_subst2
tff(fact_62_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C3: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C3))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F3,C3))) ) ) ) ) ).

% order_subst1
tff(fact_63_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_64_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3))
        <=> ! [X4: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,G3,X4))) ) ) ).

% le_fun_def
tff(fact_65_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( ! [X3: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)))
         => pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3)) ) ) ).

% le_funI
tff(fact_66_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X))) ) ) ).

% le_funE
tff(fact_67_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X))) ) ) ).

% le_funD
tff(fact_68_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
           => ( A3 = B2 ) ) ) ) ).

% antisym
tff(fact_69_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).

% dual_order.trans
tff(fact_70_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( A3 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_71_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% dual_order.eq_iff
tff(fact_72_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,fun(A,bool)),A3: A,B2: A] :
          ( ! [A5: A,B4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),P2,A5),B4)) )
         => ( ! [A5: A,B4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),P2,B4),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),P2,A5),B4)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),P2,A3),B2)) ) ) ) ).

% linorder_wlog
tff(fact_73_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2)) ) ) ) ).

% order_trans
tff(fact_74_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% order.trans
tff(fact_75_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
           => ( X = Y ) ) ) ) ).

% order_antisym
tff(fact_76_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( ( B2 = C3 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% ord_le_eq_trans
tff(fact_77_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( A3 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% ord_eq_le_trans
tff(fact_78_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).

% order_class.order_eq_iff
tff(fact_79_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) )
         => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y)) )
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
               => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2))
                   => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X)) )
                 => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X))
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_80_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & ( B2 != A3 ) ) ) ) ).

% nle_le
tff(fact_81_bounded__Max__nat,axiom,
    ! [P2: fun(nat,bool),X: nat,M5: nat] :
      ( pp(aa(nat,bool,P2,X))
     => ( ! [X3: nat] :
            ( pp(aa(nat,bool,P2,X3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),M5)) )
       => ~ ! [M: nat] :
              ( pp(aa(nat,bool,P2,M))
             => ~ ! [X5: nat] :
                    ( pp(aa(nat,bool,P2,X5))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),M)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_82_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: fun(nat,bool),K2: nat,B2: nat] :
      ( pp(aa(nat,bool,P2,K2))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => ? [X3: nat] :
            ( pp(aa(nat,bool,P2,X3))
            & ! [Y4: nat] :
                ( pp(aa(nat,bool,P2,Y4))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),X3)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_83_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
      | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2)) ) ).

% nat_le_linear
tff(fact_84_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
       => ( M2 = N ) ) ) ).

% le_antisym
tff(fact_85_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% eq_imp_le
tff(fact_86_le0,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% le0
tff(fact_87_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A3)) ).

% bot_nat_0.extremum
tff(fact_88_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),zero_zero(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_89_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),zero_zero(nat)))
     => ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_90_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),zero_zero(nat)))
    <=> ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_91_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% less_eq_nat.simps(1)
tff(fact_92_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ? [M: nat] : N = aa(nat,nat,suc,M) ) ).

% not0_implies_Suc
tff(fact_93_Zero__not__Suc,axiom,
    ! [M2: nat] : zero_zero(nat) != aa(nat,nat,suc,M2) ).

% Zero_not_Suc
tff(fact_94_Zero__neq__Suc,axiom,
    ! [M2: nat] : zero_zero(nat) != aa(nat,nat,suc,M2) ).

% Zero_neq_Suc
tff(fact_95_Suc__neq__Zero,axiom,
    ! [M2: nat] : aa(nat,nat,suc,M2) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_96_zero__induct,axiom,
    ! [P2: fun(nat,bool),K2: nat] :
      ( pp(aa(nat,bool,P2,K2))
     => ( ! [N3: nat] :
            ( pp(aa(nat,bool,P2,aa(nat,nat,suc,N3)))
           => pp(aa(nat,bool,P2,N3)) )
       => pp(aa(nat,bool,P2,zero_zero(nat))) ) ) ).

% zero_induct
tff(fact_97_diff__induct,axiom,
    ! [P2: fun(nat,fun(nat,bool)),M2: nat,N: nat] :
      ( ! [X3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,X3),zero_zero(nat)))
     => ( ! [Y3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,zero_zero(nat)),aa(nat,nat,suc,Y3)))
       => ( ! [X3: nat,Y3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,X3),Y3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,aa(nat,nat,suc,X3)),aa(nat,nat,suc,Y3))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,M2),N)) ) ) ) ).

% diff_induct
tff(fact_98_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ~ ! [N3: nat] : X != aa(nat,nat,suc,N3) ) ).

% list_decode.cases
tff(fact_99_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero(nat) != aa(nat,nat,suc,X2) ).

% nat.distinct(1)
tff(fact_100_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_101_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_102_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa(nat,nat,suc,X2) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_103_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_104_nat__induct,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( ! [N3: nat] :
            ( pp(aa(nat,bool,P2,N3))
           => pp(aa(nat,bool,P2,aa(nat,nat,suc,N3))) )
       => pp(aa(nat,bool,P2,N)) ) ) ).

% nat_induct
tff(fact_105_case4_I12_J,axiom,
    vEBT_invar_vebt(sa,deg) ).

% case4(12)
tff(fact_106_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),zero_zero(A)))
        <=> ( N = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_107_option_Osize_I4_J,axiom,
    ! [A: $tType,X2: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_108_vebt__buildup_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_buildup.cases
tff(fact_109_dependent__nat__choice,axiom,
    ! [A: $tType,P2: fun(nat,fun(A,bool)),Q: fun(nat,fun(A,fun(A,bool)))] :
      ( ? [X_1: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P2,zero_zero(nat)),X_1))
     => ( ! [X3: A,N3: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P2,N3),X3))
           => ? [Y4: A] :
                ( pp(aa(A,bool,aa(nat,fun(A,bool),P2,aa(nat,nat,suc,N3)),Y4))
                & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N3),X3),Y4)) ) )
       => ? [F2: fun(nat,A)] :
          ! [N4: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P2,N4),aa(nat,A,F2,N4)))
            & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N4),aa(nat,A,F2,N4)),aa(nat,A,F2,aa(nat,nat,suc,N4)))) ) ) ) ).

% dependent_nat_choice
tff(fact_110_exists__least__lemma,axiom,
    ! [P2: fun(nat,bool)] :
      ( ~ pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( ? [X_1: nat] : pp(aa(nat,bool,P2,X_1))
       => ? [N3: nat] :
            ( ~ pp(aa(nat,bool,P2,N3))
            & pp(aa(nat,bool,P2,aa(nat,nat,suc,N3))) ) ) ) ).

% exists_least_lemma
tff(fact_111_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),zero_zero(A))) ) ).

% le_numeral_extra(3)
tff(fact_112_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ).

% zero_le
tff(fact_113_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: A,R: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R))
        <=> ( R = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_114_one__le__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),N)) ) ) ).

% one_le_mult_iff
tff(fact_115__C1_C_I1_J,axiom,
    sa = vEBT_Leaf(a,b) ).

% "1"(1)
tff(fact_116_valid__tree__deg__neq__0,axiom,
    ! [T2: vEBT_VEBT] : ~ vEBT_invar_vebt(T2,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_117_valid__0__not,axiom,
    ! [T2: vEBT_VEBT] : ~ vEBT_invar_vebt(T2,zero_zero(nat)) ).

% valid_0_not
tff(fact_118_Leaf__0__not,axiom,
    ! [A3: bool,B2: bool] : ~ vEBT_invar_vebt(vEBT_Leaf(A3,B2),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_119_prod__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode,X) = aa(nat,product_prod(nat,nat),nat_prod_decode,Y) )
    <=> ( X = Y ) ) ).

% prod_decode_eq
tff(fact_120_insert_H__pres__valid,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => vEBT_invar_vebt(vEBT_VEBT_insert(T2,X),N) ) ).

% insert'_pres_valid
tff(fact_121_mult__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = zero_zero(nat) )
    <=> ( ( M2 = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_122_mult__0__right,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),zero_zero(nat)) = zero_zero(nat) ).

% mult_0_right
tff(fact_123_mult__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N) )
    <=> ( ( M2 = N )
        | ( K2 = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_124_mult__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),K2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2) )
    <=> ( ( M2 = N )
        | ( K2 = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_125_prod__decode__inverse,axiom,
    ! [N: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),nat_prod_decode,N)) = N ).

% prod_decode_inverse
tff(fact_126_prod__encode__inverse,axiom,
    ! [X: product_prod(nat,nat)] : aa(nat,product_prod(nat,nat),nat_prod_decode,aa(product_prod(nat,nat),nat,nat_prod_encode,X)) = X ).

% prod_encode_inverse
tff(fact_127_one__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) )
    <=> ( ( M2 = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_128_mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M2 = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_129_mult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% mult.left_commute
tff(fact_130_mult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) ) ).

% mult.commute
tff(fact_131_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% mult.assoc
tff(fact_132_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_133_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_134_Suc__mult__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),M2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),N) )
    <=> ( M2 = N ) ) ).

% Suc_mult_cancel1
tff(fact_135_mult__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% mult_0
tff(fact_136_mult__le__mono2,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),J2))) ) ).

% mult_le_mono2
tff(fact_137_mult__le__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),K2))) ) ).

% mult_le_mono1
tff(fact_138_mult__le__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),L))) ) ) ).

% mult_le_mono
tff(fact_139_le__square,axiom,
    ! [M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),M2))) ).

% le_square
tff(fact_140_le__cube,axiom,
    ! [M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),M2)))) ).

% le_cube
tff(fact_141_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_142_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_143_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: A] :
          ( ( zero_zero(A) = X )
        <=> ( X = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_144_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A3: A,C3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_145_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_146_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_147_mult__zero__right,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% mult_zero_right
tff(fact_148_mult__zero__left,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% mult_zero_left
tff(fact_149_mul__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),Y) = Z2 )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),X)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z2) ) ) ).

% mul_shift
tff(fact_150_invar__vebt_Ointros_I1_J,axiom,
    ! [A3: bool,B2: bool] : vEBT_invar_vebt(vEBT_Leaf(A3,B2),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_151_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_152_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_153_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A))) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_154_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_155_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_156_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X22: bool,Y21: bool,Y22: bool] :
      ( ( vEBT_Leaf(X21,X22) = vEBT_Leaf(Y21,Y22) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X22)
        <=> pp(Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_157_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V2048590022279873568_shift(nat,times_times(nat)) ).

% mul_def
tff(fact_158_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X22: bool] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf(X21,X22)) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_159_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) != zero_zero(A) )
         => ( ( A3 != zero_zero(A) )
            & ( B2 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_160_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = zero_zero(A) )
         => ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_161_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_162_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
          <=> ( A3 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_163_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
          <=> ( A3 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_164_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_mono
tff(fact_165_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_mono'
tff(fact_166_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3))) ) ).

% zero_le_square
tff(fact_167_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ).

% split_mult_pos_le
tff(fact_168_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).

% mult_left_mono_neg
tff(fact_169_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_170_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).

% mult_left_mono
tff(fact_171_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ) ).

% mult_right_mono_neg
tff(fact_172_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ) ).

% mult_right_mono
tff(fact_173_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).

% mult_le_0_iff
tff(fact_174_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ).

% split_mult_neg_le
tff(fact_175_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_176_valid__eq,axiom,
    ! [T2: vEBT_VEBT,D3: nat] :
      ( vEBT_VEBT_valid(T2,D3)
    <=> vEBT_invar_vebt(T2,D3) ) ).

% valid_eq
tff(fact_177_valid__eq1,axiom,
    ! [T2: vEBT_VEBT,D3: nat] :
      ( vEBT_invar_vebt(T2,D3)
     => vEBT_VEBT_valid(T2,D3) ) ).

% valid_eq1
tff(fact_178_valid__eq2,axiom,
    ! [T2: vEBT_VEBT,D3: nat] :
      ( vEBT_VEBT_valid(T2,D3)
     => vEBT_invar_vebt(T2,D3) ) ).

% valid_eq2
tff(fact_179_deg1Leaf,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,one_one(nat))
    <=> ? [A7: bool,B5: bool] : T2 = vEBT_Leaf(A7,B5) ) ).

% deg1Leaf
tff(fact_180_deg__1__Leaf,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,one_one(nat))
     => ? [A5: bool,B4: bool] : T2 = vEBT_Leaf(A5,B4) ) ).

% deg_1_Leaf
tff(fact_181_deg__1__Leafy,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( N = one_one(nat) )
       => ? [A5: bool,B4: bool] : T2 = vEBT_Leaf(A5,B4) ) ) ).

% deg_1_Leafy
tff(fact_182_nat__mult__eq__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N) )
    <=> ( ( K2 = zero_zero(nat) )
        | ( M2 = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_183_deg__not__0,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% deg_not_0
tff(fact_184_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X22: bool] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf(X21,X22)) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_185_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,N)))
     => ? [Info: option(product_prod(nat,nat)),TreeList: list(vEBT_VEBT),S: vEBT_VEBT] : Tree = vEBT_Node(Info,aa(nat,nat,suc,aa(nat,nat,suc,N)),TreeList,S) ) ).

% deg_SUcn_Node
tff(fact_186_case4_I13_J,axiom,
    vEBT_VEBT_set_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)),deg,treeList2,summary2)) = vEBT_VEBT_set_vebt(sa) ).

% case4(13)
tff(fact_187_deg__deg__n,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary),N)
     => ( Deg = N ) ) ).

% deg_deg_n
tff(fact_188_VEBT_Oinject_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,Y11: option(product_prod(nat,nat)),Y12: nat,Y13: list(vEBT_VEBT),Y14: vEBT_VEBT] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_189_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
        <=> ( N = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_190_mult__1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).

% mult_1
tff(fact_191_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).

% mult.right_neutral
tff(fact_192_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% Suc_less_eq
tff(fact_193_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N))) ) ).

% Suc_mono
tff(fact_194_lessI,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,N))) ).

% lessI
tff(fact_195_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A3)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_196_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% neq0_conv
tff(fact_197_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% less_nat_zero_code
tff(fact_198_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) )
    <=> ( ( M2 = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_199_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = one_one(nat) )
    <=> ( ( M2 = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_200_case4_I3_J,axiom,
    vEBT_invar_vebt(summary2,m) ).

% case4(3)
tff(fact_201_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,B2: A] :
          ( ( C3 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
        <=> ( ( C3 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_202_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) = C3 )
        <=> ( ( C3 = zero_zero(A) )
            | ( A3 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_203_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,B2: A] :
          ( ( C3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_204_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = C3 )
        <=> ( ( C3 = zero_zero(A) )
            | ( A3 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_205_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,N))) ).

% zero_less_Suc
tff(fact_206_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( N = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_207_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_208_mult__less__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% mult_less_cancel2
tff(fact_209_nat__0__less__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_210_less__one,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),one_one(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% less_one
tff(fact_211_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_212_mult__le__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% mult_le_cancel2
tff(fact_213_set__vebt__set__vebt_H__valid,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( vEBT_set_vebt(T2) = vEBT_VEBT_set_vebt(T2) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_214_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_215_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] :
        ? [Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X)) ) ).

% lt_ex
tff(fact_216_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] :
        ? [X_12: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X_12)) ) ).

% gt_ex
tff(fact_217_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ? [Z: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),Y)) ) ) ) ).

% dense
tff(fact_218_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( X != Y ) ) ) ).

% less_imp_neq
tff(fact_219_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% order.asym
tff(fact_220_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( A3 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).

% ord_eq_less_trans
tff(fact_221_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( ( B2 = C3 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).

% ord_less_eq_trans
tff(fact_222_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),A3: A] :
          ( ! [X3: A] :
              ( ! [Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X3))
                 => pp(aa(A,bool,P2,Y4)) )
             => pp(aa(A,bool,P2,X3)) )
         => pp(aa(A,bool,P2,A3)) ) ) ).

% less_induct
tff(fact_223_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv3
tff(fact_224_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ( X != Y )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_cases
tff(fact_225_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% dual_order.asym
tff(fact_226_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),A3)) ) ).

% dual_order.irrefl
tff(fact_227_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool)] :
          ( ? [X_13: A] : pp(aa(A,bool,P2,X_13))
        <=> ? [N5: A] :
              ( pp(aa(A,bool,P2,N5))
              & ! [M6: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M6),N5))
                 => ~ pp(aa(A,bool,P2,M6)) ) ) ) ) ).

% exists_least_iff
tff(fact_228_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,fun(A,bool)),A3: A,B2: A] :
          ( ! [A5: A,B4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A5),B4))
             => pp(aa(A,bool,aa(A,fun(A,bool),P2,A5),B4)) )
         => ( ! [A5: A] : pp(aa(A,bool,aa(A,fun(A,bool),P2,A5),A5))
           => ( ! [A5: A,B4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),P2,B4),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),P2,A5),B4)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),P2,A3),B2)) ) ) ) ) ).

% linorder_less_wlog
tff(fact_229_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).

% order.strict_trans
tff(fact_230_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
            | ( X = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_231_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).

% dual_order.strict_trans
tff(fact_232_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( A3 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_233_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( A3 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_234_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2)) ) ) ).

% nat_neq_iff
tff(fact_235_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).

% less_not_refl
tff(fact_236_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
     => ( M2 != N ) ) ).

% less_not_refl2
tff(fact_237_less__not__refl3,axiom,
    ! [S2: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S2),T2))
     => ( S2 != T2 ) ) ).

% less_not_refl3
tff(fact_238_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P2: fun(A,bool),A3: A] :
          ( ! [X3: A] :
              ( ! [Y4: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,Y4)),aa(A,B,F3,X3)))
                 => pp(aa(A,bool,P2,Y4)) )
             => pp(aa(A,bool,P2,X3)) )
         => pp(aa(A,bool,P2,A3)) ) ) ).

% measure_induct
tff(fact_239_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).

% less_irrefl_nat
tff(fact_240_nat__less__induct,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N3))
             => pp(aa(nat,bool,P2,M3)) )
         => pp(aa(nat,bool,P2,N3)) )
     => pp(aa(nat,bool,P2,N)) ) ).

% nat_less_induct
tff(fact_241_infinite__descent,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( ! [N3: nat] :
          ( ~ pp(aa(nat,bool,P2,N3))
         => ? [M3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N3))
              & ~ pp(aa(nat,bool,P2,M3)) ) )
     => pp(aa(nat,bool,P2,N)) ) ).

% infinite_descent
tff(fact_242_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_243_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P2: fun(A,bool),A3: A] :
          ( ! [X3: A] :
              ( ! [Y4: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,Y4)),aa(A,B,F3,X3)))
                 => pp(aa(A,bool,P2,Y4)) )
             => pp(aa(A,bool,P2,X3)) )
         => pp(aa(A,bool,P2,A3)) ) ) ).

% measure_induct_rule
tff(fact_244_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neqE
tff(fact_245_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_asym
tff(fact_246_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neq_iff
tff(fact_247_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% order_less_asym'
tff(fact_248_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2)) ) ) ) ).

% order_less_trans
tff(fact_249_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C3: B] :
          ( ( A3 = aa(B,A,F3,B2) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C3))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,C3))) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_250_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B2: A,F3: fun(A,B),C3: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( ( aa(A,B,F3,B2) = C3 )
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,A3)),C3)) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_251_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X)) ) ).

% order_less_irrefl
tff(fact_252_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C3: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C3))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,C3))) ) ) ) ) ).

% order_less_subst1
tff(fact_253_order__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F3: fun(A,C),C3: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,B2)),C3))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,X3)),aa(A,C,F3,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,A3)),C3)) ) ) ) ) ).

% order_less_subst2
tff(fact_254_infinite__descent__measure,axiom,
    ! [A: $tType,P2: fun(A,bool),V: fun(A,nat),X: A] :
      ( ! [X3: A] :
          ( ~ pp(aa(A,bool,P2,X3))
         => ? [Y4: A] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V,Y4)),aa(A,nat,V,X3)))
              & ~ pp(aa(A,bool,P2,Y4)) ) )
     => pp(aa(A,bool,P2,X)) ) ).

% infinite_descent_measure
tff(fact_255_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_not_sym
tff(fact_256_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,P2: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
           => pp(P2) ) ) ) ).

% order_less_imp_triv
tff(fact_257_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          | ( X = Y )
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_less_linear
tff(fact_258_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( X != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_259_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( Y != X ) ) ) ).

% order_less_imp_not_eq2
tff(fact_260_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_imp_not_less
tff(fact_261_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),one_one(A))) ) ).

% less_numeral_extra(4)
tff(fact_262_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [X: A] :
          ( ( one_one(A) = X )
        <=> ( X = one_one(A) ) ) ) ).

% one_reorient
tff(fact_263_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).

% less_numeral_extra(1)
tff(fact_264_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) ) ).

% not_one_less_zero
tff(fact_265_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).

% zero_less_one
tff(fact_266_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N: nat,M2: nat] :
          ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N3)),aa(nat,A,F3,aa(nat,nat,suc,N3))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N)),aa(nat,A,F3,M2)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2)) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_267_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N: nat,N2: nat] :
          ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N3)),aa(nat,A,F3,aa(nat,nat,suc,N3))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N)),aa(nat,A,F3,N2))) ) ) ) ).

% lift_Suc_mono_less
tff(fact_268_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M2: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M2),N))) ) ) ) ).

% less_1_mult
tff(fact_269_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D3: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf(Uu,Uv),D3)
    <=> ( D3 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_270_nat__induct__non__zero,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,P2,one_one(nat)))
       => ( ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
             => ( pp(aa(nat,bool,P2,N3))
               => pp(aa(nat,bool,P2,aa(nat,nat,suc,N3))) ) )
         => pp(aa(nat,bool,P2,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_271_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( X = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_272_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_le_less_linear
tff(fact_273_order__less__le__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F3: fun(A,C),C3: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,B2)),C3))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,X3)),aa(A,C,F3,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,A3)),C3)) ) ) ) ) ).

% order_less_le_subst2
tff(fact_274_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C3: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C3))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,C3))) ) ) ) ) ).

% order_less_le_subst1
tff(fact_275_order__le__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A3: A,B2: A,F3: fun(A,C),C3: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,B2)),C3))
           => ( ! [X3: A,Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,X3)),aa(A,C,F3,Y3))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,A3)),C3)) ) ) ) ) ).

% order_le_less_subst2
tff(fact_276_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F3: fun(B,A),B2: B,C3: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C3))
           => ( ! [X3: B,Y3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F3,C3))) ) ) ) ) ).

% order_le_less_subst1
tff(fact_277_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2)) ) ) ) ).

% order_less_le_trans
tff(fact_278_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2)) ) ) ) ).

% order_le_less_trans
tff(fact_279_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% order_neq_le_trans
tff(fact_280_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( ( A3 != B2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% order_le_neq_trans
tff(fact_281_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% order_less_imp_le
tff(fact_282_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_not_less
tff(fact_283_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_not_le
tff(fact_284_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & ( X != Y ) ) ) ) ).

% order_less_le
tff(fact_285_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( X = Y ) ) ) ) ).

% order_le_less
tff(fact_286_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% dual_order.strict_implies_order
tff(fact_287_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% order.strict_implies_order
tff(fact_288_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_289_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).

% dual_order.strict_trans2
tff(fact_290_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3)) ) ) ) ).

% dual_order.strict_trans1
tff(fact_291_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_292_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
            | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_293_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ! [W: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),W))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z2)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).

% dense_le_bounded
tff(fact_294_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X))
         => ( ! [W: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),W))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),X))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),W)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).

% dense_ge_bounded
tff(fact_295_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% order.strict_iff_not
tff(fact_296_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).

% order.strict_trans2
tff(fact_297_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C3)) ) ) ) ).

% order.strict_trans1
tff(fact_298_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_299_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
            | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_300_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% not_le_imp_less
tff(fact_301_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).

% less_le_not_le
tff(fact_302_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z2: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ).

% dense_le
tff(fact_303_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z2: A,Y: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ).

% dense_ge
tff(fact_304_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv2
tff(fact_305_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv1
tff(fact_306_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            | ( A3 = B2 ) ) ) ) ).

% nless_le
tff(fact_307_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% leI
tff(fact_308_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% leD
tff(fact_309_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
        <=> ( N != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_310_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M2: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),N))
         => ( N != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_311_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N),zero_zero(A))) ) ).

% not_less_zero
tff(fact_312_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ( N != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N)) ) ) ).

% gr_zeroI
tff(fact_313_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),zero_zero(A))) ) ).

% less_numeral_extra(3)
tff(fact_314_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M2)))
      <=> ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
tff(fact_315_strict__inc__induct,axiom,
    ! [I2: nat,J2: nat,P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( ! [I3: nat] :
            ( ( J2 = aa(nat,nat,suc,I3) )
           => pp(aa(nat,bool,P2,I3)) )
       => ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J2))
             => ( pp(aa(nat,bool,P2,aa(nat,nat,suc,I3)))
               => pp(aa(nat,bool,P2,I3)) ) )
         => pp(aa(nat,bool,P2,I2)) ) ) ) ).

% strict_inc_induct
tff(fact_316_less__Suc__induct,axiom,
    ! [I2: nat,J2: nat,P2: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( ! [I3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,I3),aa(nat,nat,suc,I3)))
       => ( ! [I3: nat,J3: nat,K: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),K))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,I3),J3))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,J3),K))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,I3),K)) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,I2),J2)) ) ) ) ).

% less_Suc_induct
tff(fact_317_less__trans__Suc,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),K2)) ) ) ).

% less_trans_Suc
tff(fact_318_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% Suc_less_SucD
tff(fact_319_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M2)))
       => ( M2 = N ) ) ) ).

% less_antisym
tff(fact_320_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
    <=> ? [M7: nat] :
          ( ( M2 = aa(nat,nat,suc,M7) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M7)) ) ) ).

% Suc_less_eq2
tff(fact_321_All__less__Suc,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ! [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P2,I)) )
    <=> ( pp(aa(nat,bool,P2,N))
        & ! [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
           => pp(aa(nat,bool,P2,I)) ) ) ) ).

% All_less_Suc
tff(fact_322_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M2))) ) ).

% not_less_eq
tff(fact_323_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
        | ( M2 = N ) ) ) ).

% less_Suc_eq
tff(fact_324_Ex__less__Suc,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ? [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P2,I)) )
    <=> ( pp(aa(nat,bool,P2,N))
        | ? [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
            & pp(aa(nat,bool,P2,I)) ) ) ) ).

% Ex_less_Suc
tff(fact_325_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,suc,N))) ) ).

% less_SucI
tff(fact_326_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
       => ( M2 = N ) ) ) ).

% less_SucE
tff(fact_327_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( ( aa(nat,nat,suc,M2) != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M2)),N)) ) ) ).

% Suc_lessI
tff(fact_328_Suc__lessE,axiom,
    ! [I2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),K2))
     => ~ ! [J3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J3))
           => ( K2 != aa(nat,nat,suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_329_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M2)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% Suc_lessD
tff(fact_330_Nat_OlessE,axiom,
    ! [I2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K2))
     => ( ( K2 != aa(nat,nat,suc,I2) )
       => ~ ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J3))
             => ( K2 != aa(nat,nat,suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_331_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: bool,X22: bool] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf(X21,X22) ).

% VEBT.distinct(1)
tff(fact_332_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: list(vEBT_VEBT),X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: bool,X222: bool] : Y != vEBT_Leaf(X212,X222) ) ).

% VEBT.exhaust
tff(fact_333_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),one_one(A))) ) ).

% le_numeral_extra(4)
tff(fact_334_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_335_bot__nat__0_Oextremum__strict,axiom,
    ! [A3: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),zero_zero(nat))) ).

% bot_nat_0.extremum_strict
tff(fact_336_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% gr0I
tff(fact_337_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
    <=> ( N = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_338_not__less0,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% not_less0
tff(fact_339_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% less_zeroE
tff(fact_340_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( N != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_341_infinite__descent0,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( ! [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
           => ( ~ pp(aa(nat,bool,P2,N3))
             => ? [M3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N3))
                  & ~ pp(aa(nat,bool,P2,M3)) ) ) )
       => pp(aa(nat,bool,P2,N)) ) ) ).

% infinite_descent0
tff(fact_342_infinite__descent0__measure,axiom,
    ! [A: $tType,V: fun(A,nat),P2: fun(A,bool),X: A] :
      ( ! [X3: A] :
          ( ( aa(A,nat,V,X3) = zero_zero(nat) )
         => pp(aa(A,bool,P2,X3)) )
     => ( ! [X3: A] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,V,X3)))
           => ( ~ pp(aa(A,bool,P2,X3))
             => ? [Y4: A] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V,Y4)),aa(A,nat,V,X3)))
                  & ~ pp(aa(A,bool,P2,Y4)) ) ) )
       => pp(aa(A,bool,P2,X)) ) ) ).

% infinite_descent0_measure
tff(fact_343_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).

% mult.comm_neutral
tff(fact_344_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_345_less__mono__imp__le__mono,axiom,
    ! [F3: fun(nat,nat),I2: nat,J2: nat] :
      ( ! [I3: nat,J3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F3,I3)),aa(nat,nat,F3,J3))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,F3,I2)),aa(nat,nat,F3,J2))) ) ) ).

% less_mono_imp_le_mono
tff(fact_346_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( ( M2 != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% le_neq_implies_less
tff(fact_347_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
        | ( M2 = N ) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% less_or_eq_imp_le
tff(fact_348_le__eq__less__or__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
        | ( M2 = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_349_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% less_imp_le_nat
tff(fact_350_nat__less__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
        & ( M2 != N ) ) ) ).

% nat_less_le
tff(fact_351_nat__mult__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),N) = N ).

% nat_mult_1
tff(fact_352_nat__mult__1__right,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),one_one(nat)) = N ).

% nat_mult_1_right
tff(fact_353_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N) )
      <=> ( M2 = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_354_nat__mult__less__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_355_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),C3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3)) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_356_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_357_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),C3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3)) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_358_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_359_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),C3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3)) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_360_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_361_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),C3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3)) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_362_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_363_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_neg_neg
tff(fact_364_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)),zero_zero(A))) ) ).

% not_square_less_zero
tff(fact_365_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).

% mult_less_0_iff
tff(fact_366_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_neg_pos
tff(fact_367_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).

% mult_pos_neg
tff(fact_368_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).

% mult_pos_pos
tff(fact_369_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A))) ) ) ) ).

% mult_pos_neg2
tff(fact_370_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_371_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos
tff(fact_372_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos2
tff(fact_373_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_374_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_375_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_376_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).

% mult_strict_left_mono
tff(fact_377_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_378_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_379_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ) ).

% mult_strict_right_mono
tff(fact_380_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_381_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_382_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) ) ).

% not_one_le_zero
tff(fact_383_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_384_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).

% zero_less_one_class.zero_le_one
tff(fact_385_nat__mult__le__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_386_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,suc,N)))
    <=> ( ( M2 = zero_zero(nat) )
        | ? [J: nat] :
            ( ( M2 = aa(nat,nat,suc,J) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_387_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ? [M: nat] : N = aa(nat,nat,suc,M) ) ).

% gr0_implies_Suc
tff(fact_388_All__less__Suc2,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ! [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P2,I)) )
    <=> ( pp(aa(nat,bool,P2,zero_zero(nat)))
        & ! [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
           => pp(aa(nat,bool,P2,aa(nat,nat,suc,I))) ) ) ) ).

% All_less_Suc2
tff(fact_389_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
    <=> ? [M6: nat] : N = aa(nat,nat,suc,M6) ) ).

% gr0_conv_Suc
tff(fact_390_Ex__less__Suc2,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ? [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P2,I)) )
    <=> ( pp(aa(nat,bool,P2,zero_zero(nat)))
        | ? [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
            & pp(aa(nat,bool,P2,aa(nat,nat,suc,I))) ) ) ) ).

% Ex_less_Suc2
tff(fact_391_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,suc,N))) ) ).

% le_imp_less_Suc
tff(fact_392_less__eq__Suc__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M2)) ) ).

% less_eq_Suc_le
tff(fact_393_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% less_Suc_eq_le
tff(fact_394_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M2)))
      <=> ( N = M2 ) ) ) ).

% le_less_Suc_eq
tff(fact_395_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% Suc_le_lessD
tff(fact_396_inc__induct,axiom,
    ! [I2: nat,J2: nat,P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,P2,J2))
       => ( ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),J2))
               => ( pp(aa(nat,bool,P2,aa(nat,nat,suc,N3)))
                 => pp(aa(nat,bool,P2,N3)) ) ) )
         => pp(aa(nat,bool,P2,I2)) ) ) ) ).

% inc_induct
tff(fact_397_dec__induct,axiom,
    ! [I2: nat,J2: nat,P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,P2,I2))
       => ( ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),J2))
               => ( pp(aa(nat,bool,P2,N3))
                 => pp(aa(nat,bool,P2,aa(nat,nat,suc,N3))) ) ) )
         => pp(aa(nat,bool,P2,J2)) ) ) ) ).

% dec_induct
tff(fact_398_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% Suc_le_eq
tff(fact_399_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N)) ) ).

% Suc_leI
tff(fact_400_ex__least__nat__le,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,N))
     => ( ~ pp(aa(nat,bool,P2,zero_zero(nat)))
       => ? [K: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
            & ! [I4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),K))
               => ~ pp(aa(nat,bool,P2,I4)) )
            & pp(aa(nat,bool,P2,K)) ) ) ) ).

% ex_least_nat_le
tff(fact_401_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_402_mult__less__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),K2))) ) ) ).

% mult_less_mono1
tff(fact_403_mult__less__mono2,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),J2))) ) ) ).

% mult_less_mono2
tff(fact_404_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_405_mult__eq__self__implies__10,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) )
     => ( ( N = one_one(nat) )
        | ( M2 = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_406_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_407_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_408_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% mult_right_le_imp_le
tff(fact_409_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% mult_left_le_imp_le
tff(fact_410_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_411_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_412_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_413_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_414_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% mult_right_less_imp_less
tff(fact_415_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_416_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_417_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% mult_left_less_imp_less
tff(fact_418_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_419_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_420_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: bool,B4: bool,X3: 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(A5,B4)),X3)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: 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(Uu2,zero_zero(nat),Uv2,Uw)),Ux)
       => ~ ! [Uy: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT,X3: 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,V2),TreeList,S)),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_421_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),A3)) ) ) ) ).

% mult_left_le
tff(fact_422_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A))) ) ) ) ) ).

% mult_le_one
tff(fact_423_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),X)) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_424_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)),X)) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_425_ex__least__nat__less,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,N))
     => ( ~ pp(aa(nat,bool,P2,zero_zero(nat)))
       => ? [K: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),N))
            & ! [I4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),K))
               => ~ pp(aa(nat,bool,P2,I4)) )
            & pp(aa(nat,bool,P2,aa(nat,nat,suc,K))) ) ) ) ).

% ex_least_nat_less
tff(fact_426_n__less__n__mult__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M2))) ) ) ).

% n_less_n_mult_m
tff(fact_427_n__less__m__mult__n,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N))) ) ) ).

% n_less_m_mult_n
tff(fact_428_one__less__mult,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N))) ) ) ).

% one_less_mult
tff(fact_429_greater__shift,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X))
    <=> pp(vEBT_VEBT_greater(aa(nat,option(nat),some(nat),X),aa(nat,option(nat),some(nat),Y))) ) ).

% greater_shift
tff(fact_430_less__shift,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
    <=> pp(vEBT_VEBT_less(aa(nat,option(nat),some(nat),X),aa(nat,option(nat),some(nat),Y))) ) ).

% less_shift
tff(fact_431_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => vEBT_invar_vebt(vEBT_vebt_buildup(N),N) ) ).

% buildup_gives_valid
tff(fact_432_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [Z: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),one_one(A)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),X)),Y)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% field_le_mult_one_interval
tff(fact_433_case4_I2_J,axiom,
    ! [S2: vEBT_VEBT] :
      ( vEBT_invar_vebt(S2,m)
     => ( ( vEBT_VEBT_set_vebt(summary2) = vEBT_VEBT_set_vebt(S2) )
       => ( S2 = summary2 ) ) ) ).

% case4(2)
tff(fact_434_maxt__corr,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),X) )
       => vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T2),X) ) ) ).

% maxt_corr
tff(fact_435_maxt__sound,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T2),X)
       => ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),X) ) ) ) ).

% maxt_sound
tff(fact_436_mint__corr,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),X) )
       => vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T2),X) ) ) ).

% mint_corr
tff(fact_437_mint__sound,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T2),X)
       => ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),X) ) ) ) ).

% mint_sound
tff(fact_438_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_439_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_440_ac,axiom,
    ! [T2: vEBT_VEBT,H: nat,K2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,H)
     => ( vEBT_invar_vebt(K2,H)
       => ( ( vEBT_VEBT_set_vebt(T2) = vEBT_VEBT_set_vebt(K2) )
         => ( vEBT_vebt_mint(T2) = vEBT_vebt_mint(K2) ) ) ) ) ).

% ac
tff(fact_441_ad,axiom,
    ! [T2: vEBT_VEBT,H: nat,K2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,H)
     => ( vEBT_invar_vebt(K2,H)
       => ( ( vEBT_VEBT_set_vebt(T2) = vEBT_VEBT_set_vebt(K2) )
         => ( vEBT_vebt_maxt(T2) = vEBT_vebt_maxt(K2) ) ) ) ) ).

% ad
tff(fact_442_case4_I5_J,axiom,
    m = na ).

% case4(5)
tff(fact_443_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less(fun(A,B)),F3),G3))
        <=> ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G3))
            & ~ pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),G3),F3)) ) ) ) ).

% less_fun_def
tff(fact_444_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_445_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(2)
tff(fact_446_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% mult_less_iff1
tff(fact_447_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz)) = aa(nat,option(nat),some(nat),Ma) ).

% vebt_maxt.simps(3)
tff(fact_448_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz)) = aa(nat,option(nat),some(nat),Mi) ).

% vebt_mint.simps(3)
tff(fact_449_maxt__corr__help,axiom,
    ! [T2: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),Maxi) )
       => ( pp(aa(nat,bool,vEBT_vebt_member(T2),X))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Maxi)) ) ) ) ).

% maxt_corr_help
tff(fact_450_mint__corr__help,axiom,
    ! [T2: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),Mini) )
       => ( pp(aa(nat,bool,vEBT_vebt_member(T2),X))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mini),X)) ) ) ) ).

% mint_corr_help
tff(fact_451_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(N),X) ).

% buildup_nothing_in_leaf
tff(fact_452_maxt__member,axiom,
    ! [T2: vEBT_VEBT,N: nat,Maxi: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),Maxi) )
       => pp(aa(nat,bool,vEBT_vebt_member(T2),Maxi)) ) ) ).

% maxt_member
tff(fact_453_mint__member,axiom,
    ! [T2: vEBT_VEBT,N: nat,Maxi: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),Maxi) )
       => pp(aa(nat,bool,vEBT_vebt_member(T2),Maxi)) ) ) ).

% mint_member
tff(fact_454_aa,axiom,
    pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),aa(set(nat),set(nat),insert(nat,mi),aa(set(nat),set(nat),insert(nat,ma),bot_bot(set(nat))))),vEBT_VEBT_set_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)),deg,treeList2,summary2)))) ).

% aa
tff(fact_455_VEBT__internal_Oinsert_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: bool,B4: bool,X3: 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(A5,B4)),X3)
     => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: 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(Info,Deg2,TreeList,Summary2)),X3) ) ).

% VEBT_internal.insert'.cases
tff(fact_456_member__correct,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,vEBT_vebt_member(T2),X))
      <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X),vEBT_set_vebt(T2))) ) ) ).

% member_correct
tff(fact_457_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),A3: A,B2: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,F3),aa(A,option(A),some(A),A3)),aa(A,option(A),some(A),B2)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),F3,A3),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_458_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_459_bot__apply,axiom,
    ! [D: $tType,C: $tType] :
      ( bot(C)
     => ! [X: D] : aa(D,C,bot_bot(fun(D,C)),X) = bot_bot(C) ) ).

% bot_apply
tff(fact_460_pred__member,axiom,
    ! [T2: vEBT_VEBT,X: nat,Y: nat] :
      ( vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T2),X,Y)
    <=> ( pp(aa(nat,bool,vEBT_vebt_member(T2),Y))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X))
        & ! [Z3: nat] :
            ( ( pp(aa(nat,bool,vEBT_vebt_member(T2),Z3))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Z3),X)) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Z3),Y)) ) ) ) ).

% pred_member
tff(fact_461_succ__member,axiom,
    ! [T2: vEBT_VEBT,X: nat,Y: nat] :
      ( vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T2),X,Y)
    <=> ( pp(aa(nat,bool,vEBT_vebt_member(T2),Y))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
        & ! [Z3: nat] :
            ( ( pp(aa(nat,bool,vEBT_vebt_member(T2),Z3))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Z3)) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),Z3)) ) ) ) ).

% succ_member
tff(fact_462_case4_I6_J,axiom,
    deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),na),m) ).

% case4(6)
tff(fact_463_case4_I1_J,axiom,
    ! [X5: vEBT_VEBT] :
      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList2)))
     => ( vEBT_invar_vebt(X5,na)
        & ! [Xa: vEBT_VEBT] :
            ( vEBT_invar_vebt(Xa,na)
           => ( ( vEBT_VEBT_set_vebt(X5) = vEBT_VEBT_set_vebt(Xa) )
             => ( Xa = X5 ) ) ) ) ) ).

% case4(1)
tff(fact_464_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X5: A] : aa(A,B,bot_bot(fun(A,B)),X5) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_465_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),bot_bot(A)))
         => ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_466_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),bot_bot(A)))
        <=> ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_467_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),bot_bot(A)),A3)) ) ).

% bot.extremum
tff(fact_468_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),bot_bot(A))) ) ).

% bot.extremum_strict
tff(fact_469_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( ( A3 != bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),A3)) ) ) ).

% bot.not_eq_extremum
tff(fact_470_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : ~ vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero(nat),Uv,Uw2),Ux2) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_471_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A3: bool,B2: bool,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf(A3,B2),X)
    <=> ( ( ( X = zero_zero(nat) )
         => pp(A3) )
        & ( ( X != zero_zero(nat) )
         => ( ( ( X = one_one(nat) )
             => pp(B2) )
            & ( X = one_one(nat) ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_472_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(Tree,N)
     => ( pp(aa(nat,bool,vEBT_vebt_member(Tree),X))
       => ( vEBT_V5719532721284313246member(Tree,X)
          | vEBT_VEBT_membermima(Tree,X) ) ) ) ).

% member_valid_both_member_options
tff(fact_473_vebt__member_Osimps_I4_J,axiom,
    ! [V3: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,X: nat] : ~ pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),X)) ).

% vebt_member.simps(4)
tff(fact_474_vebt__member_Osimps_I1_J,axiom,
    ! [A3: bool,B2: bool,X: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_Leaf(A3,B2)),X))
    <=> ( ( ( X = zero_zero(nat) )
         => pp(A3) )
        & ( ( X != zero_zero(nat) )
         => ( ( ( X = one_one(nat) )
             => pp(B2) )
            & ( X = one_one(nat) ) ) ) ) ) ).

% vebt_member.simps(1)
tff(fact_475_vebt__member_Osimps_I3_J,axiom,
    ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT,X: nat] : ~ pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz)),X)) ).

% vebt_member.simps(3)
tff(fact_476_maxt__corr__help__empty,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_maxt(T2) = none(nat) )
       => ( vEBT_VEBT_set_vebt(T2) = bot_bot(set(nat)) ) ) ) ).

% maxt_corr_help_empty
tff(fact_477_mint__corr__help__empty,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_mint(T2) = none(nat) )
       => ( vEBT_VEBT_set_vebt(T2) = bot_bot(set(nat)) ) ) ) ).

% mint_corr_help_empty
tff(fact_478_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(N),X) ).

% buildup_nothing_in_min_max
tff(fact_479_dele__member__cont__corr,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_vebt_delete(T2,X)),Y))
      <=> ( ( X != Y )
          & pp(aa(nat,bool,vEBT_vebt_member(T2),Y)) ) ) ) ).

% dele_member_cont_corr
tff(fact_480_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool,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(Uu2,Uv2)),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,Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va2,Vb2)),X3)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList,Vc2)),X3)
           => ~ ! [V2: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT,X3: 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,V2),TreeList,Vd)),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_481_case4_I8_J,axiom,
    ( ( mi = ma )
   => ! [X5: vEBT_VEBT] :
        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList2)))
       => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X5),X_1)) ) ) ).

% case4(8)
tff(fact_482_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P2: fun(A,bool),K2: A,F3: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P2,K2))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y3)),B2)) )
       => ? [X3: A] :
            ( pp(aa(A,bool,P2,X3))
            & ! [Y4: A] :
                ( pp(aa(A,bool,P2,Y4))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,Y4)),aa(A,nat,F3,X3))) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_483_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N3: nat] : X != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),N3)
     => ~ ! [N3: nat] : X != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),aa(nat,nat,suc,N3)) ) ).

% even_odd_cases
tff(fact_484_delete__pres__valid,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => vEBT_invar_vebt(vEBT_vebt_delete(T2,X),N) ) ).

% delete_pres_valid
tff(fact_485_maxbmo,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),X) )
     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X)) ) ).

% maxbmo
tff(fact_486_dele__bmo__cont__corr,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_vebt_delete(T2,X)),Y))
      <=> ( ( X != Y )
          & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),Y)) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_487_both__member__options__equiv__member,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X))
      <=> pp(aa(nat,bool,vEBT_vebt_member(T2),X)) ) ) ).

% both_member_options_equiv_member
tff(fact_488_valid__member__both__member__options,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X))
       => pp(aa(nat,bool,vEBT_vebt_member(T2),X)) ) ) ).

% valid_member_both_member_options
tff(fact_489_both__member__options__def,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X))
    <=> ( vEBT_V5719532721284313246member(T2,X)
        | vEBT_VEBT_membermima(T2,X) ) ) ).

% both_member_options_def
tff(fact_490_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) )
        <=> ( B2 = C3 ) ) ) ).

% add_right_cancel
tff(fact_491_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
        <=> ( B2 = C3 ) ) ) ).

% add_left_cancel
tff(fact_492_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),Deg)
     => ( ( Mi = Ma )
       => ( ! [X5: vEBT_VEBT] :
              ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
             => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X5),X_1)) )
          & ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_1)) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_493_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_cancel_left
tff(fact_494_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_cancel_right
tff(fact_495_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% add.right_neutral
tff(fact_496_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_497_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = A3 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_498_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = A3 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_499_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_500_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_501_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_502_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_503_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% add_0
tff(fact_504_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_cancel_right
tff(fact_505_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_cancel_left
tff(fact_506_add__Suc__right,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) ).

% add_Suc_right
tff(fact_507_Nat_Oadd__0__right,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),zero_zero(nat)) = M2 ).

% Nat.add_0_right
tff(fact_508_add__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = zero_zero(nat) )
    <=> ( ( M2 = zero_zero(nat) )
        & ( N = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_509_nat__add__left__cancel__less,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% nat_add_left_cancel_less
tff(fact_510_nat__add__left__cancel__le,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% nat_add_left_cancel_le
tff(fact_511_not__Some__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ! [Y5: A] : X != aa(A,option(A),some(A),Y5)
    <=> ( X = none(A) ) ) ).

% not_Some_eq
tff(fact_512_not__None__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ( X != none(A) )
    <=> ? [Y5: A] : X = aa(A,option(A),some(A),Y5) ) ).

% not_None_eq
tff(fact_513_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% add_le_same_cancel1
tff(fact_514_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% add_le_same_cancel2
tff(fact_515_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel1
tff(fact_516_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel2
tff(fact_517_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_518_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_519_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_520_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_521_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel2
tff(fact_522_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel1
tff(fact_523_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% add_less_same_cancel2
tff(fact_524_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% add_less_same_cancel1
tff(fact_525_add__gr__0,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% add_gr_0
tff(fact_526_mult__Suc__right,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)) ).

% mult_Suc_right
tff(fact_527_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) )
         => ( B2 = C3 ) ) ) ).

% add_right_imp_eq
tff(fact_528_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
         => ( B2 = C3 ) ) ) ).

% add_left_imp_eq
tff(fact_529_add_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% add.left_commute
tff(fact_530_add_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) ) ).

% add.commute
tff(fact_531_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) )
        <=> ( B2 = C3 ) ) ) ).

% add.right_cancel
tff(fact_532_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
        <=> ( B2 = C3 ) ) ) ).

% add.left_cancel
tff(fact_533_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% add.assoc
tff(fact_534_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B6: A,K2: A,B2: A,A3: A] :
          ( ( B6 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_535_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: A,K2: A,A3: A,B2: A] :
          ( ( A6 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),A3) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_536_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( ( I2 = J2 )
            & ( K2 = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_537_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% is_num_normalize(1)
tff(fact_538_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_539_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_540_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_541_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),J2))
            & ( K2 = L ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_542_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( ( I2 = J2 )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K2),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_543_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),J2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K2),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_544_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).

% add_mono
tff(fact_545_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))) ) ) ).

% add_left_mono
tff(fact_546_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ~ ! [C2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ) ).

% less_eqE
tff(fact_547_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))) ) ) ).

% add_right_mono
tff(fact_548_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ? [C4: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C4) ) ) ).

% le_iff_add
tff(fact_549_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_imp_le_left
tff(fact_550_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% add_le_imp_le_right
tff(fact_551_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% comm_monoid_add_class.add_0
tff(fact_552_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% add.comm_neutral
tff(fact_553_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% add.group_left_neutral
tff(fact_554_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_imp_less_right
tff(fact_555_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% add_less_imp_less_left
tff(fact_556_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))) ) ) ).

% add_strict_right_mono
tff(fact_557_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))) ) ) ).

% add_strict_left_mono
tff(fact_558_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).

% add_strict_mono
tff(fact_559_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),J2))
            & ( K2 = L ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_560_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( ( I2 = J2 )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_561_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),J2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_562_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% ring_class.ring_distribs(2)
tff(fact_563_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ).

% ring_class.ring_distribs(1)
tff(fact_564_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% comm_semiring_class.distrib
tff(fact_565_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ).

% distrib_left
tff(fact_566_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% distrib_right
tff(fact_567_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,E3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),E3)),C3) ) ).

% combine_common_factor
tff(fact_568_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,X: option(A),P2: fun(option(A),fun(option(B),bool)),Y: option(B)] :
      ( ( ( X = none(A) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P2,X),Y)) )
     => ( ( ( Y = none(B) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P2,X),Y)) )
       => ( ! [A5: A,B4: B] :
              ( ( X = aa(A,option(A),some(A),A5) )
             => ( ( Y = aa(B,option(B),some(B),B4) )
               => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P2,X),Y)) ) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P2,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_569_split__option__all,axiom,
    ! [A: $tType,P2: fun(option(A),bool)] :
      ( ! [X_13: option(A)] : pp(aa(option(A),bool,P2,X_13))
    <=> ( pp(aa(option(A),bool,P2,none(A)))
        & ! [X4: A] : pp(aa(option(A),bool,P2,aa(A,option(A),some(A),X4))) ) ) ).

% split_option_all
tff(fact_570_split__option__ex,axiom,
    ! [A: $tType,P2: fun(option(A),bool)] :
      ( ? [X_13: option(A)] : pp(aa(option(A),bool,P2,X_13))
    <=> ( pp(aa(option(A),bool,P2,none(A)))
        | ? [X4: A] : pp(aa(option(A),bool,P2,aa(A,option(A),some(A),X4))) ) ) ).

% split_option_ex
tff(fact_571_option_Oexhaust,axiom,
    ! [A: $tType,Y: option(A)] :
      ( ( Y != none(A) )
     => ~ ! [X23: A] : Y != aa(A,option(A),some(A),X23) ) ).

% option.exhaust
tff(fact_572_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X2: A] :
      ( ( Option = aa(A,option(A),some(A),X2) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_573_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X2: A] : none(A) != aa(A,option(A),some(A),X2) ).

% option.distinct(1)
tff(fact_574_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,A)),Uv2: option(A)] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv2))
     => ( ! [Uw: fun(A,fun(A,A)),V2: A] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Uw),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V2)),none(A)))
       => ~ ! [F2: fun(A,fun(A,A)),A5: A,B4: A] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),F2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),A5)),aa(A,option(A),some(A),B4))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_575_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,bool)),Uv2: option(A)] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv2))
     => ( ! [Uw: fun(A,fun(A,bool)),V2: A] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),Uw),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V2)),none(A)))
       => ~ ! [F2: fun(A,fun(A,bool)),X3: A,Y3: A] : X != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),F2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),X3)),aa(A,option(A),some(A),Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_576_nat__arith_Osuc1,axiom,
    ! [A6: nat,K2: nat,A3: nat] :
      ( ( A6 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),A3) )
     => ( aa(nat,nat,suc,A6) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(nat,nat,suc,A3)) ) ) ).

% nat_arith.suc1
tff(fact_577_add__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M2)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) ).

% add_Suc
tff(fact_578_add__Suc__shift,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M2)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,suc,N)) ).

% add_Suc_shift
tff(fact_579_add__eq__self__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = M2 )
     => ( N = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_580_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N) = N ).

% plus_nat.add_0
tff(fact_581_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),L))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% less_add_eq_less
tff(fact_582_trans__less__add2,axiom,
    ! [I2: nat,J2: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),J2))) ) ).

% trans_less_add2
tff(fact_583_trans__less__add1,axiom,
    ! [I2: nat,J2: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),M2))) ) ).

% trans_less_add1
tff(fact_584_add__less__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2))) ) ).

% add_less_mono1
tff(fact_585_not__add__less2,axiom,
    ! [J2: nat,I2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),I2)),I2)) ).

% not_add_less2
tff(fact_586_not__add__less1,axiom,
    ! [I2: nat,J2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2)),I2)) ).

% not_add_less1
tff(fact_587_add__less__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),L))) ) ) ).

% add_less_mono
tff(fact_588_add__lessD1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2)),K2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K2)) ) ).

% add_lessD1
tff(fact_589_add__leE,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2)),N))
     => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N)) ) ) ).

% add_leE
tff(fact_590_le__add1,axiom,
    ! [N: nat,M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2))) ).

% le_add1
tff(fact_591_le__add2,axiom,
    ! [N: nat,M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) ).

% le_add2
tff(fact_592_add__leD1,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% add_leD1
tff(fact_593_add__leD2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N)) ) ).

% add_leD2
tff(fact_594_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),L))
     => ? [N3: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N3) ) ).

% le_Suc_ex
tff(fact_595_add__le__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),L))) ) ) ).

% add_le_mono
tff(fact_596_add__le__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2))) ) ).

% add_le_mono1
tff(fact_597_trans__le__add1,axiom,
    ! [I2: nat,J2: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),M2))) ) ).

% trans_le_add1
tff(fact_598_trans__le__add2,axiom,
    ! [I2: nat,J2: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),J2))) ) ).

% trans_le_add2
tff(fact_599_nat__le__iff__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
    <=> ? [K3: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K3) ) ).

% nat_le_iff_add
tff(fact_600_add__mult__distrib2,axiom,
    ! [K2: nat,M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)) ).

% add_mult_distrib2
tff(fact_601_add__mult__distrib,axiom,
    ! [M2: nat,N: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2)) ).

% add_mult_distrib
tff(fact_602_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J2: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2)),U)),K2) ).

% left_add_mult_distrib
tff(fact_603_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P2: fun(A,bool),K2: A,F3: fun(A,nat),N: nat] :
      ( pp(aa(A,bool,P2,K2))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
           => ? [Y4: A] :
                ( pp(aa(A,bool,P2,Y4))
                & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,Y4)),aa(A,nat,F3,X3))) ) )
       => ? [Y3: A] :
            ( pp(aa(A,bool,P2,Y3))
            & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,K2)),N))) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_604_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw2: nat] : ~ vEBT_VEBT_membermima(vEBT_Leaf(Uu,Uv),Uw2) ).

% VEBT_internal.membermima.simps(1)
tff(fact_605_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B2)) ) ) ) ).

% add_decreasing
tff(fact_606_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).

% add_increasing
tff(fact_607_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B2)) ) ) ) ).

% add_decreasing2
tff(fact_608_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).

% add_increasing2
tff(fact_609_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_nonneg_nonneg
tff(fact_610_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_nonpos
tff(fact_611_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_612_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_613_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).

% add_less_le_mono
tff(fact_614_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).

% add_le_less_mono
tff(fact_615_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),J2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K2),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_616_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J2: A,K2: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),J2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),L))) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_617_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A))) ) ) ) ).

% add_less_zeroD
tff(fact_618_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).

% pos_add_strict
tff(fact_619_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ ! [C2: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
               => ( C2 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_620_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_pos_pos
tff(fact_621_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_neg_neg
tff(fact_622_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),one_one(A)))) ) ) ).

% add_mono1
tff(fact_623_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)))) ) ).

% less_add_one
tff(fact_624_add__is__1,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M2 = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M2 = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_625_one__is__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) )
    <=> ( ( ( M2 = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M2 = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_626_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ~ ! [Q3: nat] : N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),Q3)) ) ).

% less_natE
tff(fact_627_less__add__Suc1,axiom,
    ! [I2: nat,M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M2)))) ).

% less_add_Suc1
tff(fact_628_less__add__Suc2,axiom,
    ! [I2: nat,M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),I2)))) ).

% less_add_Suc2
tff(fact_629_less__iff__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
    <=> ? [K3: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K3)) ) ).

% less_iff_Suc_add
tff(fact_630_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ? [K: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K)) ) ).

% less_imp_Suc_add
tff(fact_631_less__imp__add__positive,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ? [K: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K) = J2 ) ) ) ).

% less_imp_add_positive
tff(fact_632_mono__nat__linear__lb,axiom,
    ! [F3: fun(nat,nat),M2: nat,K2: nat] :
      ( ! [M: nat,N3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F3,M)),aa(nat,nat,F3,N3))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F3,M2)),K2)),aa(nat,nat,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2)))) ) ).

% mono_nat_linear_lb
tff(fact_633_mult__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M2)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)) ).

% mult_Suc
tff(fact_634_Suc__eq__plus1,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_635_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_636_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N) ).

% Suc_eq_plus1_left
tff(fact_637_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X != vEBT_Leaf(fFalse,fFalse) )
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ( ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)
           => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_638_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [A: $tType,Uw2: fun(A,fun(A,A)),V3: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uw2),aa(A,option(A),some(A),V3)),none(A)) = none(A) ).

% VEBT_internal.option_shift.simps(2)
tff(fact_639_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,A)),Xa2: option(A),Xb: option(A),Y: option(A)] :
      ( ( aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,X),Xa2),Xb) = Y )
     => ( ( ( Xa2 = none(A) )
         => ( Y != none(A) ) )
       => ( ( ? [V2: A] : Xa2 = aa(A,option(A),some(A),V2)
           => ( ( Xb = none(A) )
             => ( Y != none(A) ) ) )
         => ~ ! [A5: A] :
                ( ( Xa2 = aa(A,option(A),some(A),A5) )
               => ! [B4: A] :
                    ( ( Xb = aa(A,option(A),some(A),B4) )
                   => ( Y != aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),X,A5),B4)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_640_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [E2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% field_le_epsilon
tff(fact_641_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_neg_nonpos
tff(fact_642_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_nonneg_pos
tff(fact_643_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_neg
tff(fact_644_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% add_pos_nonneg
tff(fact_645_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).

% add_strict_increasing
tff(fact_646_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3))) ) ) ) ).

% add_strict_increasing2
tff(fact_647_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))) ) ).

% sum_squares_ge_zero
tff(fact_648_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))) ) ).

% not_sum_squares_lt_zero
tff(fact_649_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),B2)) ) ) ).

% discrete
tff(fact_650_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ).

% zero_less_two
tff(fact_651_ex__has__least__nat,axiom,
    ! [A: $tType,P2: fun(A,bool),K2: A,M2: fun(A,nat)] :
      ( pp(aa(A,bool,P2,K2))
     => ? [X3: A] :
          ( pp(aa(A,bool,P2,X3))
          & ! [Y4: A] :
              ( pp(aa(A,bool,P2,Y4))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M2,X3)),aa(A,nat,M2,Y4))) ) ) ) ).

% ex_has_least_nat
tff(fact_652_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_653_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,A3: A,Y: A,U: A,V3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V3))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V3) = one_one(A) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V3),Y))),A3)) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_654_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( ! [A5: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A5,B4) )
           => ~ ( ( pp(A5)
                 => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                & ( ~ pp(A5)
                 => ( ( pp(B4)
                     => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                    & ( ~ pp(B4)
                     => ( Y = none(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw)
           => ( Y != none(nat) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_655_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( ! [A5: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A5,B4) )
           => ~ ( ( pp(B4)
                 => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                & ( ~ pp(B4)
                 => ( ( pp(A5)
                     => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                    & ( ~ pp(A5)
                     => ( Y = none(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw)
           => ( Y != none(nat) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_656_vebt__mint_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A5: bool,B4: bool] : X != vEBT_Leaf(A5,B4)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw)
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) ) ) ).

% vebt_mint.cases
tff(fact_657_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,A3: A,Y: A,U: A,V3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V3))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V3) = one_one(A) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V3),Y))),A3)) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_658_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: list(vEBT_VEBT),Vb: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va3,Vb),X)
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_659_vebt__member_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: bool,B4: bool,X3: 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(A5,B4)),X3)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw)),X3)
       => ( ! [V2: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy,Uz2)),X3)
         => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),X3)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),X3) ) ) ) ) ).

% vebt_member.cases
tff(fact_660_vebt__mint_Osimps_I1_J,axiom,
    ! [A3: bool,B2: bool] :
      ( ( pp(A3)
       => ( vEBT_vebt_mint(vEBT_Leaf(A3,B2)) = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
      & ( ~ pp(A3)
       => ( ( pp(B2)
           => ( vEBT_vebt_mint(vEBT_Leaf(A3,B2)) = aa(nat,option(nat),some(nat),one_one(nat)) ) )
          & ( ~ pp(B2)
           => ( vEBT_vebt_mint(vEBT_Leaf(A3,B2)) = none(nat) ) ) ) ) ) ).

% vebt_mint.simps(1)
tff(fact_661_vebt__maxt_Osimps_I1_J,axiom,
    ! [B2: bool,A3: bool] :
      ( ( pp(B2)
       => ( vEBT_vebt_maxt(vEBT_Leaf(A3,B2)) = aa(nat,option(nat),some(nat),one_one(nat)) ) )
      & ( ~ pp(B2)
       => ( ( pp(A3)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A3,B2)) = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
          & ( ~ pp(A3)
           => ( vEBT_vebt_maxt(vEBT_Leaf(A3,B2)) = none(nat) ) ) ) ) ) ).

% vebt_maxt.simps(1)
tff(fact_662_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Ma),X))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = none(nat) ) ) ) ).

% geqmaxNone
tff(fact_663_vebt__pred_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),zero_zero(nat))
     => ( ! [A5: bool,Uw: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,Uw)),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A5: bool,B4: bool,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(A5,B4)),aa(nat,nat,suc,aa(nat,nat,suc,Va)))
         => ( ! [Uy: nat,Uz2: list(vEBT_VEBT),Va2: 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,Va2)),Vb2)
           => ( ! [V2: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT,Vf: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd,Ve)),Vf)
             => ( ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT,Vj: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Vj)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),X3) ) ) ) ) ) ) ).

% vebt_pred.cases
tff(fact_664_vebt__succ_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,B4: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,B4)),zero_zero(nat))
     => ( ! [Uv2: bool,Uw: bool,N3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uv2,Uw)),aa(nat,nat,suc,N3))
       => ( ! [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT,Va2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)),Va2)
         => ( ! [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd)),Ve)
           => ( ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT,Vi: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Vi)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),X3) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_665_vebt__delete_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: bool,B4: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),zero_zero(nat))
     => ( ! [A5: bool,B4: bool] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A5: bool,B4: bool,N3: 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(A5,B4)),aa(nat,nat,suc,aa(nat,nat,suc,N3)))
         => ( ! [Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Uu2: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2)),Uu2)
           => ( ! [Mi2: nat,Ma2: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry)),X3)
             => ( ! [Mi2: nat,Ma2: nat,Tr: list(vEBT_VEBT),Sm: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm)),X3)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),X3) ) ) ) ) ) ) ).

% vebt_delete.cases
tff(fact_666_vebt__insert_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: bool,B4: bool,X3: 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(A5,B4)),X3)
     => ( ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X3: 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(Info,zero_zero(nat),Ts,S)),X3)
       => ( ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X3: 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(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S)),X3)
         => ( ! [V2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: 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,V2)),TreeList,Summary2)),X3)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),X3) ) ) ) ) ).

% vebt_insert.cases
tff(fact_667_succ__correct,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_succ(T2,X) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(T2),X,Sx) ) ) ).

% succ_correct
tff(fact_668_pred__correct,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_pred(T2,X) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(T2),X,Sx) ) ) ).

% pred_correct
tff(fact_669_pred__corr,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_pred(T2,X) = aa(nat,option(nat),some(nat),Px) )
      <=> vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T2),X,Px) ) ) ).

% pred_corr
tff(fact_670_succ__corr,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_succ(T2,X) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T2),X,Sx) ) ) ).

% succ_corr
tff(fact_671_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_672_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: bool,Uw2: bool,N: nat] : vEBT_vebt_succ(vEBT_Leaf(Uv,Uw2),aa(nat,nat,suc,N)) = none(nat) ).

% vebt_succ.simps(2)
tff(fact_673_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool] : vEBT_vebt_pred(vEBT_Leaf(Uu,Uv),zero_zero(nat)) = none(nat) ).

% vebt_pred.simps(1)
tff(fact_674_vebt__succ_Osimps_I4_J,axiom,
    ! [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT,Ve2: nat] : vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2),Ve2) = none(nat) ).

% vebt_succ.simps(4)
tff(fact_675_vebt__pred_Osimps_I5_J,axiom,
    ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT,Vf2: nat] : vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2),Vf2) = none(nat) ).

% vebt_pred.simps(5)
tff(fact_676_vebt__pred_Osimps_I6_J,axiom,
    ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT,Vj2: nat] : vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2),Vj2) = none(nat) ).

% vebt_pred.simps(6)
tff(fact_677_vebt__succ_Osimps_I5_J,axiom,
    ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: nat] : vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2),Vi2) = none(nat) ).

% vebt_succ.simps(5)
tff(fact_678_vebt__pred_Osimps_I2_J,axiom,
    ! [A3: bool,Uw2: bool] :
      ( ( pp(A3)
       => ( vEBT_vebt_pred(vEBT_Leaf(A3,Uw2),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
      & ( ~ pp(A3)
       => ( vEBT_vebt_pred(vEBT_Leaf(A3,Uw2),aa(nat,nat,suc,zero_zero(nat))) = none(nat) ) ) ) ).

% vebt_pred.simps(2)
tff(fact_679_vebt__succ_Osimps_I1_J,axiom,
    ! [B2: bool,Uu: bool] :
      ( ( pp(B2)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu,B2),zero_zero(nat)) = aa(nat,option(nat),some(nat),one_one(nat)) ) )
      & ( ~ pp(B2)
       => ( vEBT_vebt_succ(vEBT_Leaf(Uu,B2),zero_zero(nat)) = none(nat) ) ) ) ).

% vebt_succ.simps(1)
tff(fact_680_vebt__pred_Osimps_I3_J,axiom,
    ! [B2: bool,A3: bool,Va3: nat] :
      ( ( pp(B2)
       => ( vEBT_vebt_pred(vEBT_Leaf(A3,B2),aa(nat,nat,suc,aa(nat,nat,suc,Va3))) = aa(nat,option(nat),some(nat),one_one(nat)) ) )
      & ( ~ pp(B2)
       => ( ( pp(A3)
           => ( vEBT_vebt_pred(vEBT_Leaf(A3,B2),aa(nat,nat,suc,aa(nat,nat,suc,Va3))) = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
          & ( ~ pp(A3)
           => ( vEBT_vebt_pred(vEBT_Leaf(A3,B2),aa(nat,nat,suc,aa(nat,nat,suc,Va3))) = none(nat) ) ) ) ) ) ).

% vebt_pred.simps(3)
tff(fact_681_vebt__delete_Osimps_I3_J,axiom,
    ! [A3: bool,B2: bool,N: nat] : vEBT_vebt_delete(vEBT_Leaf(A3,B2),aa(nat,nat,suc,aa(nat,nat,suc,N))) = vEBT_Leaf(A3,B2) ).

% vebt_delete.simps(3)
tff(fact_682_vebt__delete_Osimps_I1_J,axiom,
    ! [A3: bool,B2: bool] : vEBT_vebt_delete(vEBT_Leaf(A3,B2),zero_zero(nat)) = vEBT_Leaf(fFalse,B2) ).

% vebt_delete.simps(1)
tff(fact_683_is__pred__in__set__def,axiom,
    ! [Xs: set(nat),X: nat,Y: nat] :
      ( vEBT_is_pred_in_set(Xs,X,Y)
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Y),Xs))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X))
        & ! [X4: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),Xs))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),X))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Y)) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_684_is__succ__in__set__def,axiom,
    ! [Xs: set(nat),X: nat,Y: nat] :
      ( vEBT_is_succ_in_set(Xs,X,Y)
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Y),Xs))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
        & ! [X4: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),Xs))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),X4))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X4)) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_685_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_686_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_687_vebt__delete_Osimps_I2_J,axiom,
    ! [A3: bool,B2: bool] : vEBT_vebt_delete(vEBT_Leaf(A3,B2),aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf(A3,fFalse) ).

% vebt_delete.simps(2)
tff(fact_688_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst2,Smry2),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst2,Smry2) ).

% vebt_delete.simps(5)
tff(fact_689_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT,X: nat] : vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) ).

% vebt_delete.simps(6)
tff(fact_690_add__def,axiom,
    vEBT_VEBT_add = vEBT_V2048590022279873568_shift(nat,plus_plus(nat)) ).

% add_def
tff(fact_691_delete__correct,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T2,X)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),vEBT_set_vebt(T2)),aa(set(nat),set(nat),insert(nat,X),bot_bot(set(nat)))) ) ) ).

% delete_correct
tff(fact_692_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_693_add__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y) = Z2 )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(nat,option(nat),some(nat),X)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z2) ) ) ).

% add_shift
tff(fact_694_delete__correct_H,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T2,X)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),vEBT_VEBT_set_vebt(T2)),aa(set(nat),set(nat),insert(nat,X),bot_bot(set(nat)))) ) ) ).

% delete_correct'
tff(fact_695_obtain__set__succ,axiom,
    ! [X: nat,Z2: nat,A6: set(nat),B6: set(nat)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Z2))
     => ( vEBT_VEBT_max_in_set(A6,Z2)
       => ( pp(aa(set(nat),bool,finite_finite2(nat),B6))
         => ( ( A6 = B6 )
           => ? [X_12: nat] : vEBT_is_succ_in_set(A6,X,X_12) ) ) ) ) ).

% obtain_set_succ
tff(fact_696_obtain__set__pred,axiom,
    ! [Z2: nat,X: nat,A6: set(nat)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Z2),X))
     => ( vEBT_VEBT_min_in_set(A6,Z2)
       => ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
         => ? [X_12: nat] : vEBT_is_pred_in_set(A6,X,X_12) ) ) ) ).

% obtain_set_pred
tff(fact_697_length__pos__if__in__set,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_pos_if_in_set
tff(fact_698_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X2: A] : aa(option(A),nat,size_option(A,X),aa(A,option(A),some(A),X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_699_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R: A,A3: A,B2: A,C3: A,D3: A] :
          ( ( R != zero_zero(A) )
         => ( ( ( A3 = B2 )
              & ( C3 != D3 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),R),C3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R),D3)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_700_set__vebt__finite,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => pp(aa(set(nat),bool,finite_finite2(nat),vEBT_VEBT_set_vebt(T2))) ) ).

% set_vebt_finite
tff(fact_701_pred__none__empty,axiom,
    ! [Xs: set(nat),A3: nat] :
      ( ~ ? [X_12: nat] : vEBT_is_pred_in_set(Xs,A3,X_12)
     => ( pp(aa(set(nat),bool,finite_finite2(nat),Xs))
       => ~ ? [X5: nat] :
              ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X5),Xs))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X5),A3)) ) ) ) ).

% pred_none_empty
tff(fact_702_succ__none__empty,axiom,
    ! [Xs: set(nat),A3: nat] :
      ( ~ ? [X_12: nat] : vEBT_is_succ_in_set(Xs,A3,X_12)
     => ( pp(aa(set(nat),bool,finite_finite2(nat),Xs))
       => ~ ? [X5: nat] :
              ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X5),Xs))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),X5)) ) ) ) ).

% succ_none_empty
tff(fact_703_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),A3) = zero_zero(A) ) ).

% diff_self
tff(fact_704_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),zero_zero(A)) = A3 ) ).

% diff_0_right
tff(fact_705_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% zero_diff
tff(fact_706_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),zero_zero(A)) = A3 ) ).

% diff_zero
tff(fact_707_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),A3) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_708_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = A3 ) ).

% add_diff_cancel_right'
tff(fact_709_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).

% add_diff_cancel_right
tff(fact_710_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),A3) = B2 ) ).

% add_diff_cancel_left'
tff(fact_711_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).

% add_diff_cancel_left
tff(fact_712_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),B2) = A3 ) ).

% diff_add_cancel
tff(fact_713_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = A3 ) ).

% add_diff_cancel
tff(fact_714_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% diff_ge_0_iff_ge
tff(fact_715_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% diff_gt_0_iff_gt
tff(fact_716_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),B2) = A3 ) ) ) ).

% le_add_diff_inverse2
tff(fact_717_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = A3 ) ) ) ).

% le_add_diff_inverse
tff(fact_718_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_719_diff__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),one_one(A)) = zero_zero(A) ) ) ).

% diff_numeral_special(9)
tff(fact_720_neq__if__length__neq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) != aa(list(A),nat,size_size(list(A)),Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_721_Ex__list__of__length,axiom,
    ! [A: $tType,N: nat] :
    ? [Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),Xs2) = N ).

% Ex_list_of_length
tff(fact_722_diff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C3) ) ).

% diff_right_commute
tff(fact_723_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D3) )
         => ( ( A3 = B2 )
          <=> ( C3 = D3 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_724_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,D3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3))) ) ) ) ).

% diff_mono
tff(fact_725_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2))) ) ) ).

% diff_left_mono
tff(fact_726_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3))) ) ) ).

% diff_right_mono
tff(fact_727_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3)) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_728_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_729_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3))) ) ) ).

% diff_strict_right_mono
tff(fact_730_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2))) ) ) ).

% diff_strict_left_mono
tff(fact_731_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3)) ) ) ) ).

% diff_eq_diff_less
tff(fact_732_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,D3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),D3),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3))) ) ) ) ).

% diff_strict_mono
tff(fact_733_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% left_diff_distrib
tff(fact_734_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ).

% right_diff_distrib
tff(fact_735_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B2: A,C3: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)) ) ).

% left_diff_distrib'
tff(fact_736_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ).

% right_diff_distrib'
tff(fact_737_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% diff_diff_eq
tff(fact_738_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C3: A,B2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2) = A3 )
         => ( C3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ) ) ).

% add_implies_diff
tff(fact_739_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_740_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B2) ) ).

% diff_add_eq
tff(fact_741_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B2) ) ).

% diff_diff_eq2
tff(fact_742_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) ) ).

% add_diff_eq
tff(fact_743_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,C3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = C3 ) ) ) ).

% eq_diff_eq
tff(fact_744_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = C3 )
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2) ) ) ) ).

% diff_eq_eq
tff(fact_745_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A6: A,K2: A,A3: A,B2: A] :
          ( ( A6 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),A3) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A6),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_746_finite__nat__set__iff__bounded,axiom,
    ! [N6: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),N6))
    <=> ? [M6: nat] :
        ! [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),N6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),M6)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_747_bounded__nat__set__is__finite,axiom,
    ! [N6: set(nat),N: nat] :
      ( ! [X3: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),N6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),N)) )
     => pp(aa(set(nat),bool,finite_finite2(nat),N6)) ) ).

% bounded_nat_set_is_finite
tff(fact_748_finite__nat__set__iff__bounded__le,axiom,
    ! [N6: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),N6))
    <=> ? [M6: nat] :
        ! [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),N6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),M6)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_749_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A))) ) ) ).

% le_iff_diff_le_0
tff(fact_750_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A))) ) ) ).

% less_iff_diff_less_0
tff(fact_751_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: A,K2: A,N: A,J2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K2)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K2)),J2)) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_752_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: A,K2: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K2))) ) ) ).

% add_le_imp_le_diff
tff(fact_753_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) = C3 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_754_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_755_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_756_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),C3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_757_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_758_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_759_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_760_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),B2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_761_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A3))) ) ) ).

% le_add_diff
tff(fact_762_diff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),A3) = B2 ) ) ) ).

% diff_add
tff(fact_763_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3)) ) ) ).

% le_diff_eq
tff(fact_764_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))) ) ) ).

% diff_le_eq
tff(fact_765_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3)) ) ) ).

% less_diff_eq
tff(fact_766_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))) ) ) ).

% diff_less_eq
tff(fact_767_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = A3 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_768_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),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,aa(A,fun(A,A),minus_minus(A),X),Y)) ) ).

% square_diff_square_factored
tff(fact_769_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
        <=> ( C3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E3)),D3) ) ) ) ).

% eq_add_iff2
tff(fact_770_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E3)),C3) = D3 ) ) ) ).

% eq_add_iff1
tff(fact_771_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E3)),D3))) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_772_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E3)),C3)),D3)) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_773_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E3)),C3)),D3)) ) ) ).

% less_add_iff1
tff(fact_774_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E3)),D3))) ) ) ).

% less_add_iff2
tff(fact_775_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] : aa(A,A,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_776_finite__ranking__induct,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [S3: set(B),P2: fun(set(B),bool),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S3))
         => ( pp(aa(set(B),bool,P2,bot_bot(set(B))))
           => ( ! [X3: B,S4: set(B)] :
                  ( pp(aa(set(B),bool,finite_finite2(B),S4))
                 => ( ! [Y4: B] :
                        ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y4),S4))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,Y4)),aa(B,A,F3,X3))) )
                   => ( pp(aa(set(B),bool,P2,S4))
                     => pp(aa(set(B),bool,P2,aa(set(B),set(B),insert(B,X3),S4))) ) ) )
             => pp(aa(set(B),bool,P2,S3)) ) ) ) ) ).

% finite_ranking_induct
tff(fact_777_length__induct,axiom,
    ! [A: $tType,P2: fun(list(A),bool),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys2: list(A)] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),nat,size_size(list(A)),Xs2)))
             => pp(aa(list(A),bool,P2,Ys2)) )
         => pp(aa(list(A),bool,P2,Xs2)) )
     => pp(aa(list(A),bool,P2,Xs)) ) ).

% length_induct
tff(fact_778_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B2: A,A3: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_779_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W2: A,Y: A,X: A,Z2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) )
        <=> ( ( W2 = X )
            | ( Y = Z2 ) ) ) ) ).

% crossproduct_eq
tff(fact_780_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( ( A3 != B2 )
            & ( C3 != D3 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% crossproduct_noteq
tff(fact_781_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : aa(option(A),nat,size_option(A,X),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_782_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A6))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa),X3))
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_783_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A6))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa))
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_784_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [X: A,Y: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),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,aa(A,fun(A,A),minus_minus(A),Y),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3)),B2)) ) ).

% mult_diff_mult
tff(fact_785_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% diff_shunt_var
tff(fact_786_minNullmin,axiom,
    ! [T2: vEBT_VEBT] :
      ( pp(vEBT_VEBT_minNull(T2))
     => ( vEBT_vebt_mint(T2) = none(nat) ) ) ).

% minNullmin
tff(fact_787_minminNull,axiom,
    ! [T2: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T2) = none(nat) )
     => pp(vEBT_VEBT_minNull(T2)) ) ).

% minminNull
tff(fact_788_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(X) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_maxt_rel,X)
       => ( ! [A5: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A5,B4) )
             => ( ( ( pp(B4)
                   => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                  & ( ~ pp(B4)
                   => ( ( pp(A5)
                       => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                      & ( ~ pp(A5)
                       => ( Y = none(nat) ) ) ) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Leaf(A5,B4)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw) )
               => ( ( Y = none(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Ma2) )
                   => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_789_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(X) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_mint_rel,X)
       => ( ! [A5: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A5,B4) )
             => ( ( ( pp(A5)
                   => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                  & ( ~ pp(A5)
                   => ( ( pp(B4)
                       => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                      & ( ~ pp(B4)
                       => ( Y = none(nat) ) ) ) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Leaf(A5,B4)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw) )
               => ( ( Y = none(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Mi2) )
                   => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_790_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S3))
    <=> ! [M6: nat] :
        ? [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M6),N5))
          & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N5),S3)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_791_inthall,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,bool),N: nat] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P2,X3)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),N))) ) ) ).

% inthall
tff(fact_792_not__min__Null__member,axiom,
    ! [T2: vEBT_VEBT] :
      ( ~ pp(vEBT_VEBT_minNull(T2))
     => ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X_12)) ) ).

% not_min_Null_member
tff(fact_793_min__Null__member,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( pp(vEBT_VEBT_minNull(T2))
     => ~ pp(aa(nat,bool,vEBT_vebt_member(T2),X)) ) ).

% min_Null_member
tff(fact_794_Suc__diff__diff,axiom,
    ! [M2: nat,N: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M2)),N)),aa(nat,nat,suc,K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),K2) ).

% Suc_diff_diff
tff(fact_795_diff__Suc__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) ).

% diff_Suc_Suc
tff(fact_796_diff__0__eq__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_797_diff__self__eq__0,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),M2) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_798_diff__diff__cancel,axiom,
    ! [I2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_799_diff__diff__left,axiom,
    ! [I2: nat,J2: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2)) ).

% diff_diff_left
tff(fact_800_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% zero_less_diff
tff(fact_801_diff__is__0__eq_H,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_802_diff__is__0__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% diff_is_0_eq
tff(fact_803_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2)),K2) ) ) ).

% Nat.add_diff_assoc
tff(fact_804_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)),I2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),I2)),K2) ) ) ).

% Nat.add_diff_assoc2
tff(fact_805_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),J2) ) ) ).

% Nat.diff_diff_right
tff(fact_806_diff__Suc__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,N)),one_one(nat)) = N ).

% diff_Suc_1
tff(fact_807_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% Suc_pred
tff(fact_808_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,suc,J2)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_809_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2))),I2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,J2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_810_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = N ) ) ).

% Suc_diff_1
tff(fact_811_zero__induct__lemma,axiom,
    ! [P2: fun(nat,bool),K2: nat,I2: nat] :
      ( pp(aa(nat,bool,P2,K2))
     => ( ! [N3: nat] :
            ( pp(aa(nat,bool,P2,aa(nat,nat,suc,N3)))
           => pp(aa(nat,bool,P2,N3)) )
       => pp(aa(nat,bool,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),I2))) ) ) ).

% zero_induct_lemma
tff(fact_812_minus__nat_Odiff__0,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),zero_zero(nat)) = M2 ).

% minus_nat.diff_0
tff(fact_813_diffs0__imp__equal,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2) = zero_zero(nat) )
       => ( M2 = N ) ) ) ).

% diffs0_imp_equal
tff(fact_814_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M2))) ) ) ).

% diff_less_mono2
tff(fact_815_less__imp__diff__less,axiom,
    ! [J2: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),K2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),N)),K2)) ) ).

% less_imp_diff_less
tff(fact_816_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M2))) ) ).

% diff_le_mono2
tff(fact_817_le__diff__iff_H,axiom,
    ! [A3: nat,C3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),C3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),C3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C3),A3)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C3),B2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),A3)) ) ) ) ).

% le_diff_iff'
tff(fact_818_diff__le__self,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),M2)) ).

% diff_le_self
tff(fact_819_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),L))) ) ).

% diff_le_mono
tff(fact_820_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_821_le__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ).

% le_diff_iff
tff(fact_822_eq__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2) )
        <=> ( M2 = N ) ) ) ) ).

% eq_diff_iff
tff(fact_823_Nat_Odiff__cancel,axiom,
    ! [K2: nat,M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) ).

% Nat.diff_cancel
tff(fact_824_diff__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) ).

% diff_cancel2
tff(fact_825_diff__add__inverse,axiom,
    ! [N: nat,M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),N) = M2 ).

% diff_add_inverse
tff(fact_826_diff__add__inverse2,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),N) = M2 ).

% diff_add_inverse2
tff(fact_827_diff__mult__distrib,axiom,
    ! [M2: nat,N: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2)) ).

% diff_mult_distrib
tff(fact_828_diff__mult__distrib2,axiom,
    ! [K2: nat,M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)) ).

% diff_mult_distrib2
tff(fact_829_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    pp(vEBT_VEBT_minNull(vEBT_Leaf(fFalse,fFalse))) ).

% VEBT_internal.minNull.simps(1)
tff(fact_830_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: bool] : ~ pp(vEBT_VEBT_minNull(vEBT_Leaf(fTrue,Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_831_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: bool] : ~ pp(vEBT_VEBT_minNull(vEBT_Leaf(Uu,fTrue))) ).

% VEBT_internal.minNull.simps(3)
tff(fact_832_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Ys),I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_833_Skolem__list__nth,axiom,
    ! [A: $tType,K2: nat,P2: fun(nat,fun(A,bool))] :
      ( ! [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K2))
         => ? [X_13: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P2,I),X_13)) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K2 )
          & ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K2))
             => pp(aa(A,bool,aa(nat,fun(A,bool),P2,I),aa(nat,A,nth(A,Xs3),I))) ) ) ) ).

% Skolem_list_nth
tff(fact_834_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_835_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),aa(nat,nat,suc,M2))) ).

% diff_less_Suc
tff(fact_836_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N) ) ) ).

% Suc_diff_Suc
tff(fact_837_diff__less,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),M2)) ) ) ).

% diff_less
tff(fact_838_Suc__diff__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M2)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) ) ) ).

% Suc_diff_le
tff(fact_839_less__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ) ).

% less_diff_iff
tff(fact_840_diff__less__mono,axiom,
    ! [A3: nat,B2: nat,C3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C3),A3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),C3)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C3))) ) ) ).

% diff_less_mono
tff(fact_841_diff__add__0,axiom,
    ! [N: nat,M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)) = zero_zero(nat) ).

% diff_add_0
tff(fact_842_finite__maxlen,axiom,
    ! [A: $tType,M5: set(list(A))] :
      ( pp(aa(set(list(A)),bool,finite_finite2(list(A)),M5))
     => ? [N3: nat] :
        ! [X5: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X5),M5))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X5)),N3)) ) ) ).

% finite_maxlen
tff(fact_843_less__diff__conv,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),J2)) ) ).

% less_diff_conv
tff(fact_844_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) = M2 ) ) ).

% add_diff_inverse_nat
tff(fact_845_le__diff__conv,axiom,
    ! [J2: nat,K2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)),I2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2))) ) ).

% le_diff_conv
tff(fact_846_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),J2)) ) ) ).

% Nat.le_diff_conv2
tff(fact_847_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)) ) ) ).

% Nat.diff_add_assoc
tff(fact_848_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),I2)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_849_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2) = K2 )
      <=> ( J2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_850_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))),N) ).

% diff_Suc_eq_diff_pred
tff(fact_851_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va3: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : ~ pp(vEBT_VEBT_minNull(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va3,Vb,Vc))) ).

% VEBT_internal.minNull.simps(5)
tff(fact_852_nth__mem,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,nth(A,Xs),N)),aa(list(A),set(A),set2(A),Xs))) ) ).

% nth_mem
tff(fact_853_list__ball__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A),P2: fun(A,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P2,X3)) )
       => pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),N))) ) ) ).

% list_ball_nth
tff(fact_854_in__set__conv__nth,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
    <=> ? [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( aa(nat,A,nth(A,Xs),I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_855_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,bool),X: A] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),I3))) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => pp(aa(A,bool,P2,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_856_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P2,X4)) )
    <=> ! [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),I))) ) ) ).

% all_set_conv_all_nth
tff(fact_857_diff__Suc__less,axiom,
    ! [N: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,I2))),N)) ) ).

% diff_Suc_less
tff(fact_858_nat__diff__split,axiom,
    ! [P2: fun(nat,bool),A3: nat,B2: nat] :
      ( pp(aa(nat,bool,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
         => pp(aa(nat,bool,P2,zero_zero(nat))) )
        & ! [D4: nat] :
            ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
           => pp(aa(nat,bool,P2,D4)) ) ) ) ).

% nat_diff_split
tff(fact_859_nat__diff__split__asm,axiom,
    ! [P2: fun(nat,bool),A3: nat,B2: nat] :
      ( pp(aa(nat,bool,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)))
    <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
            & ~ pp(aa(nat,bool,P2,zero_zero(nat))) )
          | ? [D4: nat] :
              ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
              & ~ pp(aa(nat,bool,P2,D4)) ) ) ) ).

% nat_diff_split_asm
tff(fact_860_less__diff__conv2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),K2)),I2))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2))) ) ) ).

% less_diff_conv2
tff(fact_861_nat__eq__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),U)),M2) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_862_nat__eq__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N) )
      <=> ( M2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_863_nat__le__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),U)),M2)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_864_nat__le__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_865_nat__diff__add__eq1,axiom,
    ! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),U)),M2)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_866_nat__diff__add__eq2,axiom,
    ! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_867_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(vEBT_VEBT_minNull(X))
     => ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
       => ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
         => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_868_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(vEBT_VEBT_minNull(X))
     => ( ( X != vEBT_Leaf(fFalse,fFalse) )
       => ~ ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_869_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_870_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M2)),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_871_add__eq__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = N ) )
      & ( ( M2 != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))),N)) ) ) ) ).

% add_eq_if
tff(fact_872_nat__less__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),U)),M2)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_873_nat__less__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J2),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_874_mult__eq__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = zero_zero(nat) ) )
      & ( ( M2 != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat))),N)) ) ) ) ).

% mult_eq_if
tff(fact_875_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(vEBT_VEBT_minNull(X))
      <=> pp(Y) )
     => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
         => ~ pp(Y) )
       => ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
           => pp(Y) )
         => ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
             => pp(Y) )
           => ( ( ? [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)
               => ~ pp(Y) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2)
                 => pp(Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_876_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D1))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D22))
           => ? [E2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),E2),D1))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),E2),D22)) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_877_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A3))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A6))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa),X3))
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_878_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A6))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa))
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_879_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: bool] :
      ( ( pp(vEBT_VEBT_minNull(X))
      <=> pp(Y) )
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,X)
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ( pp(Y)
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fFalse,fFalse)) ) )
         => ( ! [Uv2: bool] :
                ( ( X = vEBT_Leaf(fTrue,Uv2) )
               => ( ~ pp(Y)
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fTrue,Uv2)) ) )
           => ( ! [Uu2: bool] :
                  ( ( X = vEBT_Leaf(Uu2,fTrue) )
                 => ( ~ pp(Y)
                   => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(Uu2,fTrue)) ) )
             => ( ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy) )
                   => ( pp(Y)
                     => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) )
                     => ( ~ pp(Y)
                       => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_880_Suc__if__eq,axiom,
    ! [A: $tType,F3: fun(nat,A),H: fun(nat,A),G3: A,N: nat] :
      ( ! [N3: nat] : aa(nat,A,F3,aa(nat,nat,suc,N3)) = aa(nat,A,H,N3)
     => ( ( aa(nat,A,F3,zero_zero(nat)) = G3 )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,F3,N) = G3 ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,F3,N) = aa(nat,A,H,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ) ) ).

% Suc_if_eq
tff(fact_881_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( pp(vEBT_VEBT_minNull(X))
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,X)
       => ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
           => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fFalse,fFalse)) )
         => ~ ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_882_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ pp(vEBT_VEBT_minNull(X))
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,X)
       => ( ! [Uv2: bool] :
              ( ( X = vEBT_Leaf(fTrue,Uv2) )
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(fTrue,Uv2)) )
         => ( ! [Uu2: bool] :
                ( ( X = vEBT_Leaf(Uu2,fTrue) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf(Uu2,fTrue)) )
           => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) )
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_883_nth__enumerate__eq,axiom,
    ! [A: $tType,M2: nat,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,N,Xs)),M2) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)),aa(nat,A,nth(A,Xs),M2)) ) ) ).

% nth_enumerate_eq
tff(fact_884_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S3: set(A),Y: A,F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S3))
         => ( ( S3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),S3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S3))),aa(A,B,F3,Y))) ) ) ) ) ).

% arg_min_least
tff(fact_885_Euclid__induct,axiom,
    ! [P2: fun(nat,fun(nat,bool)),A3: nat,B2: nat] :
      ( ! [A5: nat,B4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,A5),B4))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,B4),A5)) )
     => ( ! [A5: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,A5),zero_zero(nat)))
       => ( ! [A5: nat,B4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,A5),B4))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,A5),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),B4))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,A3),B2)) ) ) ) ).

% Euclid_induct
tff(fact_886_nat__descend__induct,axiom,
    ! [N: nat,P2: fun(nat,bool),M2: nat] :
      ( ! [K: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
         => pp(aa(nat,bool,P2,K)) )
     => ( ! [K: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => ( ! [I4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),I4))
                 => pp(aa(nat,bool,P2,I4)) )
             => pp(aa(nat,bool,P2,K)) ) )
       => pp(aa(nat,bool,P2,M2)) ) ) ).

% nat_descend_induct
tff(fact_887_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ! [X3: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
         => ( aa(list(A),nat,size_size(list(A)),X3) = N ) )
     => ( aa(list(A),nat,size_size(list(A)),concat(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(list(A)),nat,size_size(list(list(A))),Xs)),N) ) ) ).

% length_mul_elem
tff(fact_888_triangle__Suc,axiom,
    ! [N: nat] : nat_triangle(aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(N)),aa(nat,nat,suc,N)) ).

% triangle_Suc
tff(fact_889_length__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,N,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_enumerate
tff(fact_890_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_891_diff__commute,axiom,
    ! [I2: nat,J2: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J2)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),K2)),J2) ).

% diff_commute
tff(fact_892_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(nat,nat)] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(X),Xa2) = Y )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
         => ( Y = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa2)) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
         => ( Y = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa2),aa(nat,nat,suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_893_prod__decode__aux_Osimps,axiom,
    ! [M2: nat,K2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),K2))
       => ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K2),M2) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),M2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),K2))
       => ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K2),M2) = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,K2))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_894_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,M2: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K2),M2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K2)),M2) ).

% prod_encode_prod_decode_aux
tff(fact_895_prod__decode__triangle__add,axiom,
    ! [K2: nat,M2: nat] : aa(nat,product_prod(nat,nat),nat_prod_decode,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K2)),M2)) = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K2),M2) ).

% prod_decode_triangle_add
tff(fact_896_in__set__product__lists__length,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),aa(list(list(A)),set(list(A)),set2(list(A)),product_lists(A,Xss))))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Xss) ) ) ).

% in_set_product_lists_length
tff(fact_897_Gcd__remove0__nat,axiom,
    ! [M5: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),M5))
     => ( gcd_Gcd(nat,M5) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M5),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_898_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P2: fun(A,bool),D5: A,Q: fun(A,bool)] :
          ( ! [X3: A,K: A] :
              ( pp(aa(A,bool,P2,X3))
            <=> pp(aa(A,bool,P2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K),D5)))) )
         => ( ! [X3: A,K: A] :
                ( pp(aa(A,bool,Q,X3))
              <=> pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K),D5)))) )
           => ! [X5: A,K4: A] :
                ( ( pp(aa(A,bool,P2,X5))
                  | pp(aa(A,bool,Q,X5)) )
              <=> ( pp(aa(A,bool,P2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))))
                  | pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5)))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_899_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P2: fun(A,bool),D5: A,Q: fun(A,bool)] :
          ( ! [X3: A,K: A] :
              ( pp(aa(A,bool,P2,X3))
            <=> pp(aa(A,bool,P2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K),D5)))) )
         => ( ! [X3: A,K: A] :
                ( pp(aa(A,bool,Q,X3))
              <=> pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K),D5)))) )
           => ! [X5: A,K4: A] :
                ( ( pp(aa(A,bool,P2,X5))
                  & pp(aa(A,bool,Q,X5)) )
              <=> ( pp(aa(A,bool,P2,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))))
                  & pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5)))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_900_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% verit_sum_simplify
tff(fact_901_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_902_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A6: set(A)] :
          ( ( gcd_Gcd(A,A6) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),insert(A,zero_zero(A)),bot_bot(set(A))))) ) ) ).

% Gcd_0_iff
tff(fact_903_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% verit_la_disequality
tff(fact_904_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),A3)) ) ).

% verit_comp_simplify1(2)
tff(fact_905_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero(nat)) ).

% prod_decode_def
tff(fact_906_verit__comp__simplify1_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B3: B,A4: B] :
          ( ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B3),A4))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A4),B3)) ) ) ).

% verit_comp_simplify1(3)
tff(fact_907_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X5))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),T2)) ) ) ).

% pinf(6)
tff(fact_908_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X5))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),X5)) ) ) ).

% pinf(8)
tff(fact_909_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),T2)) ) ) ).

% minf(6)
tff(fact_910_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z: A] :
        ! [X5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Z))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),X5)) ) ) ).

% minf(8)
tff(fact_911_length__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),product(A,B,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).

% length_product
tff(fact_912_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A3: A,B2: A,P2: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,P2,A3))
           => ( ~ pp(aa(A,bool,P2,B2))
             => ? [C2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
                  & ! [X5: A] :
                      ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X5))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),C2)) )
                     => pp(aa(A,bool,P2,X5)) )
                  & ! [D6: A] :
                      ( ! [X3: A] :
                          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),D6)) )
                         => pp(aa(A,bool,P2,X3)) )
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D6),C2)) ) ) ) ) ) ) ).

% complete_interval
tff(fact_913_vebt__insert_Osimps_I4_J,axiom,
    ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),X)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary) ).

% vebt_insert.simps(4)
tff(fact_914_nth__zip,axiom,
    ! [A: $tType,B: $tType,I2: nat,Xs: list(A),Ys: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Ys)))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys)),I2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I2)),aa(nat,B,nth(B,Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_915_find__Some__iff2,axiom,
    ! [A: $tType,X: A,P2: fun(A,bool),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),X) = find(A,P2,Xs) )
    <=> ? [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),I)))
          & ( X = aa(nat,A,nth(A,Xs),I) )
          & ! [J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),I))
             => ~ pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),J))) ) ) ) ).

% find_Some_iff2
tff(fact_916_find__Some__iff,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A),X: A] :
      ( ( find(A,P2,Xs) = aa(A,option(A),some(A),X) )
    <=> ? [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),I)))
          & ( X = aa(nat,A,nth(A,Xs),I) )
          & ! [J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),I))
             => ~ pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),J))) ) ) ) ).

% find_Some_iff
tff(fact_917_power_Opower__eq__if,axiom,
    ! [A: $tType,M2: nat,One: A,Times: fun(A,fun(A,A)),P: A] :
      ( ( ( M2 = zero_zero(nat) )
       => ( power2(A,One,Times,P,M2) = One ) )
      & ( ( M2 != zero_zero(nat) )
       => ( power2(A,One,Times,P,M2) = aa(A,A,aa(A,fun(A,A),Times,P),power2(A,One,Times,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat)))) ) ) ) ).

% power.power_eq_if
tff(fact_918_nth__Cons__pos,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_919_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( rotate1(A,Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_920_length__rotate1,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),rotate1(A,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rotate1
tff(fact_921_nth__Cons__Suc,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(nat,nat,suc,N)) = aa(nat,A,nth(A,Xs),N) ).

% nth_Cons_Suc
tff(fact_922_nth__Cons__0,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),zero_zero(nat)) = X ).

% nth_Cons_0
tff(fact_923_zip__Cons__Cons,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),zip(A,B,Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_924_enumerate__simps_I2_J,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] : enumerate(B,N,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(product_prod(nat,B)),list(product_prod(nat,B)),aa(product_prod(nat,B),fun(list(product_prod(nat,B)),list(product_prod(nat,B))),cons(product_prod(nat,B)),aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),N),X)),enumerate(B,aa(nat,nat,suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_925_zip__eq__ConsE,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xy: product_prod(A,B),Xys: list(product_prod(A,B))] :
      ( ( zip(A,B,Xs,Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),Xy),Xys) )
     => ~ ! [X3: A,Xs4: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
           => ! [Y3: B,Ys3: list(B)] :
                ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3) )
               => ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) )
                 => ( Xys != zip(A,B,Xs4,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_926_find_Osimps_I2_J,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P2,X))
       => ( find(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,option(A),some(A),X) ) )
      & ( ~ pp(aa(A,bool,P2,X))
       => ( find(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = find(A,P2,Xs) ) ) ) ).

% find.simps(2)
tff(fact_927_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),aa(list(B),set(B),set2(B),Ys))) ) ).

% set_zip_rightD
tff(fact_928_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% set_zip_leftD
tff(fact_929_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
     => ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
         => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),aa(list(B),set(B),set2(B),Ys))) ) ) ).

% in_set_zipE
tff(fact_930_zip__same,axiom,
    ! [A: $tType,A3: A,B2: A,Xs: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Xs))))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),aa(list(A),set(A),set2(A),Xs)))
        & ( A3 = B2 ) ) ) ).

% zip_same
tff(fact_931_Suc__length__conv,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( aa(nat,nat,suc,N) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ? [Y5: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_932_length__Suc__conv,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y5: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_933_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)))
     => ( Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys) ) ) ).

% impossible_Cons
tff(fact_934_power_Opower_Opower__Suc,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A3: A,N: nat] : power2(A,One,Times,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),Times,A3),power2(A,One,Times,A3,N)) ).

% power.power.power_Suc
tff(fact_935_power_Opower_Opower__0,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A3: A] : power2(A,One,Times,A3,zero_zero(nat)) = One ).

% power.power.power_0
tff(fact_936_vebt__insert_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info2,zero_zero(nat),Ts2,S2),X) = vEBT_Node(Info2,zero_zero(nat),Ts2,S2) ).

% vebt_insert.simps(2)
tff(fact_937_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ~ ! [Y3: B] : ~ pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_938_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),aa(list(B),set(B),set2(B),Ys)))
       => ~ ! [X3: A] : ~ pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_939_Suc__le__length__iff,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ? [X4: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys4) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_940_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
    ! [A3: bool,B2: bool,X: nat] : vEBT_VEBT_insert(vEBT_Leaf(A3,B2),X) = vEBT_vebt_insert(vEBT_Leaf(A3,B2),X) ).

% VEBT_internal.insert'.simps(1)
tff(fact_941_vebt__insert_Osimps_I3_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2),X) = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) ).

% vebt_insert.simps(3)
tff(fact_942_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A3: bool,B2: bool] :
      ( ( ( X = zero_zero(nat) )
       => ( vEBT_vebt_insert(vEBT_Leaf(A3,B2),X) = vEBT_Leaf(fTrue,B2) ) )
      & ( ( X != zero_zero(nat) )
       => ( ( ( X = one_one(nat) )
           => ( vEBT_vebt_insert(vEBT_Leaf(A3,B2),X) = vEBT_Leaf(A3,fTrue) ) )
          & ( ( X != one_one(nat) )
           => ( vEBT_vebt_insert(vEBT_Leaf(A3,B2),X) = vEBT_Leaf(A3,B2) ) ) ) ) ) ).

% vebt_insert.simps(1)
tff(fact_943_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X22: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_944_nth__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).

% nth_Cons'
tff(fact_945_nth__equal__first__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X )
        <=> ( N = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_946_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A),N: nat] :
      ( ( X != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = Y )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_947_length__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_948_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_949_enumerate__Suc_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] : infini527867602293511546merate(A,S3,aa(nat,nat,suc,N)) = infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,infini527867602293511546merate(A,S3,zero_zero(nat))),bot_bot(set(A)))),N) ) ).

% enumerate_Suc'
tff(fact_950_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_dec(A,X) = aa(A,A,aa(A,fun(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_951_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(A)] :
          ( ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A6) = zero_zero(A) )
        <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),insert(A,zero_zero(A)),bot_bot(set(A)))))
            & pp(aa(set(A),bool,finite_finite2(A),A6)) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_952_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2)) ) ) ) ) ).

% power_decreasing_iff
tff(fact_953_card__insert__le__m1,axiom,
    ! [A: $tType,N: nat,Y: set(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_954_count__notin,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(A,nat,count_list(A,Xs),X) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_955_Cons__lenlex__iff,axiom,
    ! [A: $tType,M2: A,Ms: list(A),N: A,Ns: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M2),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N),Ns))),lenlex(A,R)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M2),N)),R)) )
        | ( ( M2 = N )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R))) ) ) ) ).

% Cons_lenlex_iff
tff(fact_956_power__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),Y) = Z2 )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_power,aa(nat,option(nat),some(nat),X)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z2) ) ) ).

% power_shift
tff(fact_957_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V2048590022279873568_shift(nat,power_power(nat)) ).

% local.power_def
tff(fact_958_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),M2) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M2 = zero_zero(nat) )
        | ( X = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_959_power__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_960_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
        | ( N = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_961_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_962_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(nat,nat,suc,N)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_963_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% power_Suc0_right
tff(fact_964_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_965_card_Oinfinite,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( aa(set(A),nat,finite_card(A),A6) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_966_Gcd__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_fin.empty
tff(fact_967_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% power_eq_0_iff
tff(fact_968_card__0__eq,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( aa(set(A),nat,finite_card(A),A6) = zero_zero(nat) )
      <=> ( A6 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_969_card__insert__disjoint,axiom,
    ! [A: $tType,A6: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A6)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A6)) ) ) ) ).

% card_insert_disjoint
tff(fact_970_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2)) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_971_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y)) ) ) ) ).

% power_increasing_iff
tff(fact_972_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ) ).

% power_mono_iff
tff(fact_973_power__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,N: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) != zero_zero(A) ) ) ) ).

% power_not_zero
tff(fact_974_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y: A,N: 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,aa(A,fun(nat,A),power_power(A),X),N)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) ) ) ) ).

% power_commuting_commutes
tff(fact_975_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)) ) ).

% power_mult_distrib
tff(fact_976_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_commutes
tff(fact_977_power__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M2: nat,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),N) ) ).

% power_mult
tff(fact_978_finite__le__enumerate,axiom,
    ! [S3: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(set(nat),nat,finite_card(nat),S3)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),infini527867602293511546merate(nat,S3,N))) ) ) ).

% finite_le_enumerate
tff(fact_979_lenlex__irreflexive,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R))
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lenlex(A,R))) ) ).

% lenlex_irreflexive
tff(fact_980_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% zero_le_power
tff(fact_981_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono
tff(fact_982_finite__enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),S3)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S3,N)),infini527867602293511546merate(A,S3,aa(nat,nat,suc,N)))) ) ) ) ).

% finite_enumerate_step
tff(fact_983_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B6: set(A),A6: set(B),R: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),B6))
     => ( ! [A5: B] :
            ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),A6))
           => ? [B7: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B7),B6))
                & pp(aa(A,bool,aa(B,fun(A,bool),R,A5),B7)) ) )
       => ( ! [A1: B,A22: B,B4: A] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A1),A6))
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A22),A6))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B6))
                 => ( pp(aa(A,bool,aa(B,fun(A,bool),R,A1),B4))
                   => ( pp(aa(A,bool,aa(B,fun(A,bool),R,A22),B4))
                     => ( A1 = A22 ) ) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A6)),aa(set(A),nat,finite_card(A),B6))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_984_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% zero_less_power
tff(fact_985_card__insert__le,axiom,
    ! [A: $tType,A6: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A6)))) ).

% card_insert_le
tff(fact_986_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% one_le_power
tff(fact_987_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y: A,N: 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,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_988_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X6: set(A),Y6: set(A)] :
          ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(set(A),nat,finite_card(A),X6)))
             => ( infini527867602293511546merate(A,X6,I3) = infini527867602293511546merate(A,Y6,I3) ) )
         => ( pp(aa(set(A),bool,finite_finite2(A),X6))
           => ( pp(aa(set(A),bool,finite_finite2(A),Y6))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),X6)),aa(set(A),nat,finite_card(A),Y6)))
               => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),Y6)) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_989_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_Suc
tff(fact_990_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),A3) ) ).

% power_Suc2
tff(fact_991_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_992_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M2: nat,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_add
tff(fact_993_nat__power__less__imp__less,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),N)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_994_le__enumerate,axiom,
    ! [S3: set(nat),N: nat] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),S3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),infini527867602293511546merate(nat,S3,N))) ) ).

% le_enumerate
tff(fact_995_card__eq__0__iff,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A6) = zero_zero(nat) )
    <=> ( ( A6 = bot_bot(set(A)) )
        | ~ pp(aa(set(A),bool,finite_finite2(A),A6)) ) ) ).

% card_eq_0_iff
tff(fact_996_card__ge__0__finite,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A6)))
     => pp(aa(set(A),bool,finite_finite2(A),A6)) ) ).

% card_ge_0_finite
tff(fact_997_card__insert__if,axiom,
    ! [A: $tType,A6: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A6)) = aa(set(A),nat,finite_card(A),A6) ) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A6)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A6)) ) ) ) ) ).

% card_insert_if
tff(fact_998_card__Suc__eq__finite,axiom,
    ! [A: $tType,A6: set(A),K2: nat] :
      ( ( aa(set(A),nat,finite_card(A),A6) = aa(nat,nat,suc,K2) )
    <=> ? [B5: A,B8: set(A)] :
          ( ( A6 = aa(set(A),set(A),insert(A,B5),B8) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B5),B8))
          & ( aa(set(A),nat,finite_card(A),B8) = K2 )
          & pp(aa(set(A),bool,finite_finite2(A),B8)) ) ) ).

% card_Suc_eq_finite
tff(fact_999_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% power_less_imp_less_base
tff(fact_1000_card__mono,axiom,
    ! [A: $tType,B6: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B6))) ) ) ).

% card_mono
tff(fact_1001_card__seteq,axiom,
    ! [A: $tType,B6: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B6)),aa(set(A),nat,finite_card(A),A6)))
         => ( A6 = B6 ) ) ) ) ).

% card_seteq
tff(fact_1002_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F4: set(A),C5: nat] :
      ( ! [G4: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),G4),F4))
         => ( pp(aa(set(A),bool,finite_finite2(A),G4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G4)),C5)) ) )
     => ( pp(aa(set(A),bool,finite_finite2(A),F4))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F4)),C5)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_1003_obtain__subset__with__card__n,axiom,
    ! [A: $tType,N: nat,S3: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),S3)))
     => ~ ! [T3: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T3),S3))
           => ( ( aa(set(A),nat,finite_card(A),T3) = N )
             => ~ pp(aa(set(A),bool,finite_finite2(A),T3)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_1004_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A))) ) ) ) ).

% power_le_one
tff(fact_1005_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( A3 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_1006_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% power_le_imp_le_base
tff(fact_1007_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))) ) ) ).

% power_less_power_Suc
tff(fact_1008_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))) ) ) ).

% power_gt1_lemma
tff(fact_1009_card__le__sym__Diff,axiom,
    ! [A: $tType,A6: set(A),B6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,finite_finite2(A),B6))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B6)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B6))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B6),A6)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_1010_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% card_length
tff(fact_1011_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)))) ) ) ).

% power_gt1
tff(fact_1012_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ) ).

% power_0_left
tff(fact_1013_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N6: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N6))) ) ) ) ).

% power_increasing
tff(fact_1014_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_1015_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))) ) ).

% lenlex_length
tff(fact_1016_power__gt__expt,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),K2))) ) ).

% power_gt_expt
tff(fact_1017_nat__one__le__power,axiom,
    ! [I2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),N))) ) ).

% nat_one_le_power
tff(fact_1018_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,S3,N)),infini527867602293511546merate(A,S3,aa(nat,nat,suc,N)))) ) ) ).

% enumerate_step
tff(fact_1019_card__gt__0__iff,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A6)))
    <=> ( ( A6 != bot_bot(set(A)) )
        & pp(aa(set(A),bool,finite_finite2(A),A6)) ) ) ).

% card_gt_0_iff
tff(fact_1020_card__1__singleton__iff,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A6) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X4: A] : A6 = aa(set(A),set(A),insert(A,X4),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_1021_card__eq__SucD,axiom,
    ! [A: $tType,A6: set(A),K2: nat] :
      ( ( aa(set(A),nat,finite_card(A),A6) = aa(nat,nat,suc,K2) )
     => ? [B4: A,B9: set(A)] :
          ( ( A6 = aa(set(A),set(A),insert(A,B4),B9) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B9))
          & ( aa(set(A),nat,finite_card(A),B9) = K2 )
          & ( ( K2 = zero_zero(nat) )
           => ( B9 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_1022_card__Suc__eq,axiom,
    ! [A: $tType,A6: set(A),K2: nat] :
      ( ( aa(set(A),nat,finite_card(A),A6) = aa(nat,nat,suc,K2) )
    <=> ? [B5: A,B8: set(A)] :
          ( ( A6 = aa(set(A),set(A),insert(A,B5),B8) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B5),B8))
          & ( aa(set(A),nat,finite_card(A),B8) = K2 )
          & ( ( K2 = zero_zero(nat) )
           => ( B8 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_1023_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(nat,nat,suc,zero_zero(nat))))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
           => ! [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A6))
               => ( X4 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_1024_card__le__Suc__iff,axiom,
    ! [A: $tType,N: nat,A6: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),A6)))
    <=> ? [A7: A,B8: set(A)] :
          ( ( A6 = aa(set(A),set(A),insert(A,A7),B8) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A7),B8))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),B8)))
          & pp(aa(set(A),bool,finite_finite2(A),B8)) ) ) ).

% card_le_Suc_iff
tff(fact_1025_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).

% power_Suc_less
tff(fact_1026_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),A3)) ) ) ) ).

% power_Suc_le_self
tff(fact_1027_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),one_one(A))) ) ) ) ).

% power_Suc_less_one
tff(fact_1028_card__Diff1__le,axiom,
    ! [A: $tType,A6: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A6))) ).

% card_Diff1_le
tff(fact_1029_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N6: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N6)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1030_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N6: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N6)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).

% power_decreasing
tff(fact_1031_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1032_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
              <=> ( A3 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1033_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
               => ( A3 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1034_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B6: set(A),A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B6))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B6))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B6)))) ) ).

% diff_card_le_card_Diff
tff(fact_1035_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).

% self_le_power
tff(fact_1036_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).

% one_less_power
tff(fact_1037_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,count_list(A,Xs),X)),aa(list(A),nat,size_size(list(A)),Xs))) ).

% count_le_length
tff(fact_1038_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_1039_card_Oremove,axiom,
    ! [A: $tType,A6: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
       => ( aa(set(A),nat,finite_card(A),A6) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_1040_card_Oinsert__remove,axiom,
    ! [A: $tType,A6: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,X),A6)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_1041_card__Suc__Diff1,axiom,
    ! [A: $tType,A6: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A6) ) ) ) ).

% card_Suc_Diff1
tff(fact_1042_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ) ).

% power_strict_mono
tff(fact_1043_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [M2: nat,P: A] :
          ( ( ( M2 = zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P),M2) = one_one(A) ) )
          & ( ( M2 != zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),P),aa(nat,A,aa(A,fun(nat,A),power_power(A),P),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat)))) ) ) ) ) ).

% power_eq_if
tff(fact_1044_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),A3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) ) ) ).

% power_minus_mult
tff(fact_1045_Cons__in__lex,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),lex(A,R)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( X = Y )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R))) ) ) ) ).

% Cons_in_lex
tff(fact_1046_realpow__pos__nth__unique,axiom,
    ! [N: nat,A3: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
       => ? [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X3))
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X3),N) = A3 )
            & ! [Y4: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y4))
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),N) = A3 ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1047_realpow__pos__nth,axiom,
    ! [N: nat,A3: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
       => ? [R3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),N) = A3 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1048_sprop1,axiom,
    ( ( sa = vEBT_Node(info,deg,treeList,summary) )
    & ( deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),na),m) )
    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
    & vEBT_invar_vebt(summary,m)
    & ! [X5: vEBT_VEBT] :
        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList)))
       => vEBT_invar_vebt(X5,na) ) ) ).

% sprop1
tff(fact_1049_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),N5)),aa(nat,B,nth(B,Ys),N5))),R)) ) ) ) ).

% listrel_iff_nth
tff(fact_1050_in__measures_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,F3: fun(A,nat),Fs: list(fun(A,nat))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F3),Fs))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
        | ( ( aa(A,nat,F3,X) = aa(A,nat,F3,Y) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,Fs))) ) ) ) ).

% in_measures(2)
tff(fact_1051_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),X)),N)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
            | ( N = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1052__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062treeList_H_Asummary_H_Ainfo_O_As_A_061_ANode_Ainfo_Adeg_AtreeList_H_Asummary_H_A_092_060and_062_Adeg_A_061_An_A_L_Am_A_092_060and_062_Alength_AtreeList_H_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_H_Am_A_092_060and_062_A_I_092_060forall_062t_092_060in_062set_AtreeList_H_O_Ainvar__vebt_At_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [TreeList3: list(vEBT_VEBT),Summary3: vEBT_VEBT,Info: option(product_prod(nat,nat))] :
        ~ ( ( sa = vEBT_Node(Info,deg,TreeList3,Summary3) )
          & ( deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),na),m) )
          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
          & vEBT_invar_vebt(Summary3,m)
          & ! [X5: vEBT_VEBT] :
              ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
             => vEBT_invar_vebt(X5,na) ) ) ).

% \<open>\<And>thesis. (\<And>treeList' summary' info. s = Node info deg treeList' summary' \<and> deg = n + m \<and> length treeList' = 2 ^ m \<and> invar_vebt summary' m \<and> (\<forall>t\<in>set treeList'. invar_vebt t n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_1053_intind,axiom,
    ! [A: $tType,I2: nat,N: nat,P2: fun(A,bool),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
     => ( pp(aa(A,bool,P2,X))
       => pp(aa(A,bool,P2,aa(nat,A,nth(A,replicate(A,N,X)),I2))) ) ) ).

% intind
tff(fact_1054_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M2: num,N: num] :
          ( ( aa(num,A,numeral_numeral(A),M2) = aa(num,A,numeral_numeral(A),N) )
        <=> ( M2 = N ) ) ) ).

% numeral_eq_iff
tff(fact_1055_semiring__norm_I13_J,axiom,
    ! [M2: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M2)),bit0(N)) = bit0(bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M2),N))) ).

% semiring_norm(13)
tff(fact_1056_semiring__norm_I11_J,axiom,
    ! [M2: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M2),one2) = M2 ).

% semiring_norm(11)
tff(fact_1057_semiring__norm_I12_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),one2),N) = N ).

% semiring_norm(12)
tff(fact_1058_case4_I10_J,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),deg))) ).

% case4(10)
tff(fact_1059_case4_I4_J,axiom,
    aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) ).

% case4(4)
tff(fact_1060_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M2: nat,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M2) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( M2 = N ) ) ) ).

% of_nat_eq_iff
tff(fact_1061_a0,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),deg)) ).

% a0
tff(fact_1062_case4_I7_J,axiom,
    ! [I4: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m)))
     => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList2),I4)),X_13))
      <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,summary2),I4)) ) ) ).

% case4(7)
tff(fact_1063_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(Tree),X))
     => ( vEBT_invar_vebt(Tree,N)
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ) ).

% member_bound
tff(fact_1064_valid__pres__insert,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
       => vEBT_invar_vebt(vEBT_vebt_insert(T2,X),N) ) ) ).

% valid_pres_insert
tff(fact_1065_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M2),N)) ) ) ).

% numeral_le_iff
tff(fact_1066_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N)) ) ) ).

% numeral_less_iff
tff(fact_1067_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)) ) ).

% numeral_times_numeral
tff(fact_1068_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V3: num,W2: num,Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V3)),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),V3),W2))),Z2) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_1069_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V3: num,W2: num,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V3)),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),V3),W2))),Z2) ) ).

% add_numeral_left
tff(fact_1070_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)) ) ).

% numeral_plus_numeral
tff(fact_1071_num__double,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(one2)),N) = bit0(N) ).

% num_double
tff(fact_1072_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M2: num,N: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),M2))),aa(num,nat,numeral_numeral(nat),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N))) ) ).

% power_mult_numeral
tff(fact_1073_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_1074_misiz,axiom,
    ! [T2: vEBT_VEBT,N: nat,M2: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( aa(nat,option(nat),some(nat),M2) = vEBT_vebt_mint(T2) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ) ).

% misiz
tff(fact_1075_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),X),X)))
    <=> ( X = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_1076_valid__insert__both__member__options__add,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
       => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T2,X)),X)) ) ) ).

% valid_insert_both_member_options_add
tff(fact_1077_valid__insert__both__member__options__pres,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
         => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X))
           => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T2,Y)),X)) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_1078_helpypredd,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_pred(T2,X) = aa(nat,option(nat),some(nat),Y) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ) ).

% helpypredd
tff(fact_1079_helpyd,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_succ(T2,X) = aa(nat,option(nat),some(nat),Y) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ) ).

% helpyd
tff(fact_1080_post__member__pre__member,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
         => ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_vebt_insert(T2,X)),Y))
           => ( pp(aa(nat,bool,vEBT_vebt_member(T2),Y))
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_1081_replicate__eq__replicate,axiom,
    ! [A: $tType,M2: nat,X: A,N: nat,Y: A] :
      ( ( replicate(A,M2,X) = replicate(A,N,Y) )
    <=> ( ( M2 = N )
        & ( ( M2 != zero_zero(nat) )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_1082_length__replicate,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),nat,size_size(list(A)),replicate(A,N,X)) = N ).

% length_replicate
tff(fact_1083_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary) ) ) ) ).

% delt_out_of_range
tff(fact_1084_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary) ) ) ) ).

% del_single_cont
tff(fact_1085_set__n__deg__not__0,axiom,
    ! [TreeList2: list(vEBT_VEBT),N: nat,M2: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
         => vEBT_invar_vebt(X3,N) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N)) ) ) ).

% set_n_deg_not_0
tff(fact_1086_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),Ma))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_1087_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = aa(nat,option(nat),some(nat),Ma) ) ) ) ).

% pred_max
tff(fact_1088_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = aa(nat,option(nat),some(nat),Mi) ) ) ) ).

% succ_min
tff(fact_1089_bit__concat__def,axiom,
    ! [H: nat,L: nat,D3: nat] : vEBT_VEBT_bit_concat(H,L,D3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),D3))),L) ).

% bit_concat_def
tff(fact_1090_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V3: num,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V3)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V3)),C3)) ) ).

% distrib_left_numeral
tff(fact_1091_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A3: A,B2: A,V3: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(num,A,numeral_numeral(A),V3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V3))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V3))) ) ).

% distrib_right_numeral
tff(fact_1092_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] :
          ( ( aa(num,A,numeral_numeral(A),N) = one_one(A) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_1093_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),N) )
        <=> ( one2 = N ) ) ) ).

% one_eq_numeral_iff
tff(fact_1094_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A3: A,B2: A,V3: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),aa(num,A,numeral_numeral(A),V3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V3))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V3))) ) ).

% left_diff_distrib_numeral
tff(fact_1095_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V3: num,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V3)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V3)),C3)) ) ).

% right_diff_distrib_numeral
tff(fact_1096_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K2: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),K2)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_1097_Suc__numeral,axiom,
    ! [N: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),N)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).

% Suc_numeral
tff(fact_1098_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),M2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N))) ) ).

% power_add_numeral
tff(fact_1099_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M2: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),M2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)))),B2) ) ).

% power_add_numeral2
tff(fact_1100_of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M2) = zero_zero(A) )
        <=> ( M2 = zero_zero(nat) ) ) ) ).

% of_nat_eq_0_iff
tff(fact_1101_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( zero_zero(nat) = N ) ) ) ).

% of_nat_0_eq_iff
tff(fact_1102_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_1103_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% of_nat_less_iff
tff(fact_1104_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: num] : aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),N)) = aa(num,A,numeral_numeral(A),N) ) ).

% of_nat_numeral
tff(fact_1105_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% of_nat_le_iff
tff(fact_1106_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_add
tff(fact_1107_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M2: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mult
tff(fact_1108_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = one_one(A) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_1109_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_1110_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_1111_in__set__replicate,axiom,
    ! [A: $tType,X: A,N: nat,Y: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),replicate(A,N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_1112_Bex__set__replicate,axiom,
    ! [A: $tType,N: nat,A3: A,P2: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),replicate(A,N,A3))))
          & pp(aa(A,bool,P2,X4)) )
    <=> ( pp(aa(A,bool,P2,A3))
        & ( N != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_1113_Ball__set__replicate,axiom,
    ! [A: $tType,N: nat,A3: A,P2: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),replicate(A,N,A3))))
         => pp(aa(A,bool,P2,X4)) )
    <=> ( pp(aa(A,bool,P2,A3))
        | ( N = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_1114_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),one_one(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),one2)) ) ) ).

% numeral_le_one_iff
tff(fact_1115_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),N)) ) ) ).

% one_less_numeral_iff
tff(fact_1116_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% one_plus_numeral
tff(fact_1117_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ) ).

% numeral_plus_one
tff(fact_1118_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M2)),zero_zero(A)))
        <=> ( M2 = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_1119_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M2: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M2)) ) ).

% of_nat_Suc
tff(fact_1120_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_1121_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_1122_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% of_nat_0_less_iff
tff(fact_1123_add__2__eq__Suc,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc
tff(fact_1124_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc'
tff(fact_1125_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_1126_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2)),X)) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1127_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2))) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1128_set__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_1129_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
            <=> ( X = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1130_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A)))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_1131_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_1132_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(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_1133_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: num,N: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I2)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I2)),N)),X)) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1134_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I2: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I2)),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I2)),N))) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1135_power2__nat__le__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% power2_nat_le_imp_le
tff(fact_1136_power2__nat__le__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% power2_nat_le_eq_le
tff(fact_1137_self__le__ge2__pow,axiom,
    ! [K2: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K2),M2))) ) ).

% self_le_ge2_pow
tff(fact_1138_card__2__iff_H,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S3) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
          & ? [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),S3))
              & ( X4 != Xa3 )
              & ! [Xb2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xb2),S3))
                 => ( ( Xb2 = X4 )
                    | ( Xb2 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_1139_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ! [Y4: real] :
        ? [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N3)),X))) ) ).

% reals_Archimedean3
tff(fact_1140_add__One__commute,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N) = aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2) ).

% add_One_commute
tff(fact_1141_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bit0(N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_Bit0
tff(fact_1142_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),one2)) = A3 ) ).

% mult_numeral_1_right
tff(fact_1143_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A3) = A3 ) ).

% mult_numeral_1
tff(fact_1144_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_1145_pos2,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% pos2
tff(fact_1146_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_1147_le__num__One__iff,axiom,
    ! [X: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),X),one2))
    <=> ( X = one2 ) ) ).

% le_num_One_iff
tff(fact_1148_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_1149_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_1150_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),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))),A3)),B2) ) ).

% left_add_twice
tff(fact_1151_zero__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) ) ) ).

% zero_power2
tff(fact_1152_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).

% power2_eq_square
tff(fact_1153_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A] : aa(nat,A,aa(A,fun(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_1154_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_1155_power__even__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power_even_eq
tff(fact_1156_diff__le__diff__pow,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K2),N)))) ) ).

% diff_le_diff_pow
tff(fact_1157_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_1158_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,aa(A,fun(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,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(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_1159_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% sum_squares_bound
tff(fact_1160_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(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_1161_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% power2_le_imp_le
tff(fact_1162_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
             => ( X = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1163_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% zero_le_power2
tff(fact_1164_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))) ) ).

% power2_less_0
tff(fact_1165_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_1166_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_1167_card__2__iff,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S3) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X4: A,Y5: A] :
          ( ( S3 = aa(set(A),set(A),insert(A,X4),aa(set(A),set(A),insert(A,Y5),bot_bot(set(A)))) )
          & ( X4 != Y5 ) ) ) ).

% card_2_iff
tff(fact_1168_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_1169_Suc__nat__number__of__add,axiom,
    ! [V3: num,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V3)),N)) = 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),V3),one2))),N) ).

% Suc_nat_number_of_add
tff(fact_1170_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(nat,A,semiring_1_of_nat(A),N3))) ) ).

% real_arch_simple
tff(fact_1171_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,semiring_1_of_nat(A),N3))) ) ).

% reals_Archimedean2
tff(fact_1172_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_1173_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2))
    <=> ? [N5: 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,N5))) ) ).

% zless_iff_Suc_zadd
tff(fact_1174_int__Suc,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ).

% int_Suc
tff(fact_1175_int__ops_I4_J,axiom,
    ! [A3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),one_one(int)) ).

% int_ops(4)
tff(fact_1176_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),N) ) ).

% zero_neq_numeral
tff(fact_1177_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_1178_nat__le__real__less,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),M2)),one_one(real)))) ) ).

% nat_le_real_less
tff(fact_1179_zle__int,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M2)),aa(nat,int,semiring_1_of_nat(int),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% zle_int
tff(fact_1180_nat__int__comparison_I3_J,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).

% nat_int_comparison(3)
tff(fact_1181_int__ops_I7_J,axiom,
    ! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(7)
tff(fact_1182_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% power2_less_imp_less
tff(fact_1183_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(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_1184_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% sum_power2_ge_zero
tff(fact_1185_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(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_1186_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A))) ) ).

% not_sum_power2_lt_zero
tff(fact_1187_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_1188_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ) ).

% zero_le_even_power'
tff(fact_1189_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% power_odd_eq
tff(fact_1190_ex__power__ivl1,axiom,
    ! [B2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K2))
       => ? [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N3)),K2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),one_one(nat))))) ) ) ) ).

% ex_power_ivl1
tff(fact_1191_ex__power__ivl2,axiom,
    ! [B2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2))
       => ? [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N3)),K2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),one_one(nat))))) ) ) ) ).

% ex_power_ivl2
tff(fact_1192_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Deg: nat,X: nat,Info2: option(product_prod(nat,nat)),TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)),X))
       => ( vEBT_VEBT_insert(vEBT_Node(Info2,Deg,TreeList2,Summary),X) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)),X))
       => ( vEBT_VEBT_insert(vEBT_Node(Info2,Deg,TreeList2,Summary),X) = vEBT_vebt_insert(vEBT_Node(Info2,Deg,TreeList2,Summary),X) ) ) ) ).

% VEBT_internal.insert'.simps(2)
tff(fact_1193_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_1194_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),zero_zero(A))) ) ) ).

% odd_power_less_zero
tff(fact_1195_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(X,Xa2) = Y )
     => ( ! [A5: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A5,B4) )
           => ( Y != vEBT_vebt_insert(vEBT_Leaf(A5,B4),Xa2) ) )
       => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
             => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)),Xa2))
                   => ( Y = vEBT_Node(Info,Deg2,TreeList,Summary2) ) )
                  & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)),Xa2))
                   => ( Y = vEBT_vebt_insert(vEBT_Node(Info,Deg2,TreeList,Summary2),Xa2) ) ) ) ) ) ) ).

% VEBT_internal.insert'.elims
tff(fact_1196_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% of_nat_0_le_iff
tff(fact_1197_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M2)),zero_zero(A))) ) ).

% of_nat_less_0_iff
tff(fact_1198_of__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)) != zero_zero(A) ) ).

% of_nat_neq_0
tff(fact_1199_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ) ).

% of_nat_less_imp_less
tff(fact_1200_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N))) ) ) ).

% less_imp_of_nat_less
tff(fact_1201_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I2: nat,J2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I2)),aa(nat,A,semiring_1_of_nat(A),J2))) ) ) ).

% of_nat_mono
tff(fact_1202_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_le_numeral
tff(fact_1203_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_le_zero
tff(fact_1204_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_less_zero
tff(fact_1205_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_less_numeral
tff(fact_1206_replicate__Suc,axiom,
    ! [A: $tType,N: nat,X: A] : replicate(A,aa(nat,nat,suc,N),X) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),replicate(A,N,X)) ).

% replicate_Suc
tff(fact_1207_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C3))
       => ( ! [M: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),X)),C3)) )
         => ( X = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1208_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% one_le_numeral
tff(fact_1209_pos__int__cases,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
     => ~ ! [N3: nat] :
            ( ( K2 = aa(nat,int,semiring_1_of_nat(int),N3) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3)) ) ) ).

% pos_int_cases
tff(fact_1210_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
     => ? [N3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
          & ( K2 = aa(nat,int,semiring_1_of_nat(int),N3) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_1211_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J2: int,K2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),J2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K2)),I2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K2)),J2))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1212_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),one_one(A))) ) ).

% not_numeral_less_one
tff(fact_1213_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_1214_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ( X3 = X ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_1215_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
     => ( ! [Y3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),aa(list(A),set(A),set2(A),Xs)))
           => ( Y3 = X ) )
       => ( Xs = replicate(A,N,X) ) ) ) ).

% replicate_eqI
tff(fact_1216_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
         => vEBT_invar_vebt(X3,N) )
     => ( vEBT_invar_vebt(Summary,M2)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
         => ( ( M2 = N )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
             => ( ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                     => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_12)) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_1217_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
         => vEBT_invar_vebt(X3,N) )
     => ( vEBT_invar_vebt(Summary,M2)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
         => ( ( M2 = aa(nat,nat,suc,N) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
             => ( ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_12))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                     => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_12)) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_1218_listrel__eq__len,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% listrel_eq_len
tff(fact_1219_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N3)),X))) ) ) ).

% ex_less_of_nat_mult
tff(fact_1220_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [N: nat,M2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ) ).

% of_nat_diff
tff(fact_1221_zdiff__int__split,axiom,
    ! [P2: fun(int,bool),X: nat,Y: nat] :
      ( pp(aa(int,bool,P2,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Y))))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
         => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Y)))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
         => pp(aa(int,bool,P2,zero_zero(int))) ) ) ) ).

% zdiff_int_split
tff(fact_1222_num_Osize_I5_J,axiom,
    ! [X2: num] : aa(num,nat,size_size(num),bit0(X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_1223_realpow__pos__nth2,axiom,
    ! [A3: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ? [R3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
          & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),aa(nat,nat,suc,N)) = A3 ) ) ) ).

% realpow_pos_nth2
tff(fact_1224_listrel__Cons2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))),listrel(A,B,R)))
     => ~ ! [X3: A,Xs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
           => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y)),R))
             => ~ pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys)),listrel(A,B,R))) ) ) ) ).

% listrel_Cons2
tff(fact_1225_listrel__Cons1,axiom,
    ! [B: $tType,A: $tType,Y: A,Ys: list(A),Xs: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),Xs)),listrel(A,B,R)))
     => ~ ! [Y3: B,Ys5: list(B)] :
            ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5) )
           => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Y3)),R))
             => ~ pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Ys),Ys5)),listrel(A,B,R))) ) ) ) ).

% listrel_Cons1
tff(fact_1226_listrel_OCons,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,R: set(product_prod(A,B)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),R))
     => ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R)))
       => pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))),listrel(A,B,R))) ) ) ).

% listrel.Cons
tff(fact_1227_measures__less,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F3),Fs)))) ) ).

% measures_less
tff(fact_1228_measures__lesseq,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,Fs)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F3),Fs)))) ) ) ).

% measures_lesseq
tff(fact_1229_set__replicate__Suc,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,N),X)) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_1230_set__replicate__conv__if,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = bot_bot(set(A)) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ) ) ).

% set_replicate_conv_if
tff(fact_1231_Cons__replicate__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat,Y: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = replicate(A,N,Y) )
    <=> ( ( X = Y )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
        & ( Xs = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),X) ) ) ) ).

% Cons_replicate_eq
tff(fact_1232_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,R: A,Q2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q2)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R),aa(num,A,numeral_numeral(A),L))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q2)),R) ) ) ) ) ).

% divmod_step_eq
tff(fact_1233_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),N)
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),Ma))
          & ? [M: nat] :
              ( ( aa(nat,option(nat),some(nat),M) = vEBT_vebt_mint(Summary) )
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ) ) ) ).

% mintlistlength
tff(fact_1234_insert__correct,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
       => ( 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),insert(nat,X),bot_bot(set(nat)))) = vEBT_set_vebt(vEBT_vebt_insert(T2,X)) ) ) ) ).

% insert_correct
tff(fact_1235_insert__corr,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
       => ( 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),insert(nat,X),bot_bot(set(nat)))) = vEBT_VEBT_set_vebt(vEBT_vebt_insert(T2,X)) ) ) ) ).

% insert_corr
tff(fact_1236_inrange,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),vEBT_VEBT_set_vebt(T2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat))))) ) ).

% inrange
tff(fact_1237_nat__bit__induct,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( ! [N3: nat] :
            ( pp(aa(nat,bool,P2,N3))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
             => pp(aa(nat,bool,P2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3))) ) )
       => ( ! [N3: nat] :
              ( pp(aa(nat,bool,P2,N3))
             => pp(aa(nat,bool,P2,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)))) )
         => pp(aa(nat,bool,P2,N)) ) ) ) ).

% nat_bit_induct
tff(fact_1238_nat__induct2,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( pp(aa(nat,bool,P2,one_one(nat)))
       => ( ! [N3: nat] :
              ( pp(aa(nat,bool,P2,N3))
             => pp(aa(nat,bool,P2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
         => pp(aa(nat,bool,P2,N)) ) ) ) ).

% nat_induct2
tff(fact_1239_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,M2: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1240_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1241_pow__sum,axiom,
    ! [A3: nat,B2: nat] : divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A3)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2) ).

% pow_sum
tff(fact_1242_power__minus__is__div,axiom,
    ! [B2: nat,A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),A3))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)) ) ) ).

% power_minus_is_div
tff(fact_1243_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : divide_divide(A,A3,zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_1244_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : divide_divide(A,zero_zero(A),A3) = zero_zero(A) ) ).

% bits_div_0
tff(fact_1245_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : divide_divide(A,A3,zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_1246_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : divide_divide(A,zero_zero(A),A3) = zero_zero(A) ) ).

% div_0
tff(fact_1247_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : divide_divide(A,A3,zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_1248_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,C3: A,B2: A] :
          ( ( divide_divide(A,A3,C3) = divide_divide(A,B2,C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_1249_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( divide_divide(A,C3,A3) = divide_divide(A,C3,B2) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_1250_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( divide_divide(A,A3,B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_1251_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) ) ).

% times_divide_eq_right
tff(fact_1252_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A,C3: A] : divide_divide(A,A3,divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),B2) ) ).

% divide_divide_eq_right
tff(fact_1253_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A,C3: A] : divide_divide(A,divide_divide(A,A3,B2),C3) = divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% divide_divide_eq_left
tff(fact_1254_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C3: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C3)),A3) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),C3) ) ).

% times_divide_eq_left
tff(fact_1255_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : divide_divide(A,A3,one_one(A)) = A3 ) ).

% div_by_1
tff(fact_1256_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),set_or1337092689740270186AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),U)) ) ) ) ).

% atLeastAtMost_iff
tff(fact_1257_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L2: A,H2: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L2,H2) )
        <=> ( ( ( L = L2 )
              & ( H = H2 ) )
            | ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
              & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L2),H2)) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_1258_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa(set(nat),bool,finite_finite2(nat),set_or1337092689740270186AtMost(nat,L,U))) ).

% finite_atLeastAtMost
tff(fact_1259_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),A3) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1260_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),B2) = A3 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1261_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( ( C3 = zero_zero(A) )
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = zero_zero(A) ) )
          & ( ( C3 != zero_zero(A) )
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A3,B2) ) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1262_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A3,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1263_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,A3,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1264_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,A3,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1265_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A3,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1266_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( divide_divide(A,A3,A3) = one_one(A) ) ) ) ).

% div_self
tff(fact_1267_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = divide_divide(A,one_one(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_1268_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( divide_divide(A,one_one(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_1269_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( ( one_one(A) = divide_divide(A,B2,A3) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_1270_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( ( divide_divide(A,B2,A3) = one_one(A) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_1271_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( ( A3 = zero_zero(A) )
           => ( divide_divide(A,A3,A3) = zero_zero(A) ) )
          & ( ( A3 != zero_zero(A) )
           => ( divide_divide(A,A3,A3) = one_one(A) ) ) ) ) ).

% divide_self_if
tff(fact_1272_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( divide_divide(A,A3,A3) = one_one(A) ) ) ) ).

% divide_self
tff(fact_1273_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( one_one(A) = divide_divide(A,A3,B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_1274_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( divide_divide(A,A3,B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_1275_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% atLeastatMost_empty_iff
tff(fact_1276_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A3,B2) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_1277_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C3,D3)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1278_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_1279_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or1337092689740270186AtMost(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Icc_iff
tff(fact_1280_atLeastAtMost__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : set_or1337092689740270186AtMost(A,A3,A3) = aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_1281_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B2) = aa(set(A),set(A),insert(A,C3),bot_bot(set(A))) )
        <=> ( ( A3 = B2 )
            & ( B2 = C3 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_1282_nat__mult__div__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ( K2 = zero_zero(nat) )
       => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)) = zero_zero(nat) ) )
      & ( ( K2 != zero_zero(nat) )
       => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)) = divide_divide(nat,M2,N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_1283_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,one_one(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% zero_le_divide_1_iff
tff(fact_1284_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% divide_le_0_1_iff
tff(fact_1285_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,one_one(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% zero_less_divide_1_iff
tff(fact_1286_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),divide_divide(A,B2,A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_1287_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),divide_divide(A,B2,A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_1288_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_1289_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_1290_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,one_one(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% divide_less_0_1_iff
tff(fact_1291_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_1292_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))),B2)) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_1293_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W2: num,A3: A] :
          ( ( divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2)) = A3 )
        <=> ( ( ( aa(num,A,numeral_numeral(A),W2) != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)) ) )
            & ( ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1294_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,W2: num] :
          ( ( A3 = divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2)) )
        <=> ( ( ( aa(num,A,numeral_numeral(A),W2) != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)) = B2 ) )
            & ( ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1295_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))),B2)) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_1296_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_1297_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = divide_divide(A,one_one(A),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1298_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,B2,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = divide_divide(A,one_one(A),A3) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1299_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M2: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(num,real,numeral_numeral(real),N)),aa(nat,real,semiring_1_of_nat(real),M2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),N)),M2)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_1300_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or1337092689740270186AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,U)),L) ).

% card_atLeastAtMost
tff(fact_1301_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_1302_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_1303_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_1304_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,A3)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_1305_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_1306_decr__mult__lemma,axiom,
    ! [D3: int,P2: fun(int,bool),K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int] :
            ( pp(aa(int,bool,P2,X3))
           => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D3))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
         => ! [X5: int] :
              ( pp(aa(int,bool,P2,X5))
             => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X5),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1307_minusinfinity,axiom,
    ! [D3: int,P1: fun(int,bool),P2: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int,K: int] :
            ( pp(aa(int,bool,P1,X3))
          <=> pp(aa(int,bool,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3)))) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),Z4))
             => ( pp(aa(int,bool,P2,X3))
              <=> pp(aa(int,bool,P1,X3)) ) )
         => ( ? [X_1: int] : pp(aa(int,bool,P1,X_1))
           => ? [X_12: int] : pp(aa(int,bool,P2,X_12)) ) ) ) ) ).

% minusinfinity
tff(fact_1308_plusinfinity,axiom,
    ! [D3: int,P3: fun(int,bool),P2: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int,K: int] :
            ( pp(aa(int,bool,P3,X3))
          <=> pp(aa(int,bool,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3)))) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),X3))
             => ( pp(aa(int,bool,P2,X3))
              <=> pp(aa(int,bool,P3,X3)) ) )
         => ( ? [X_1: int] : pp(aa(int,bool,P3,X_1))
           => ? [X_12: int] : pp(aa(int,bool,P2,X_12)) ) ) ) ) ).

% plusinfinity
tff(fact_1309_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_1310_times__int__code_I1_J,axiom,
    ! [K2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),K2),zero_zero(int)) = zero_zero(int) ).

% times_int_code(1)
tff(fact_1311_pos__zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M2))
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M2),N) = one_one(int) )
      <=> ( ( M2 = one_one(int) )
          & ( N = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1312_zmult__zless__mono2,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),J2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I2)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),J2))) ) ) ).

% zmult_zless_mono2
tff(fact_1313_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q4: int,R4: int,Q2: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),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),Q2)),R)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R4))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q2),Q4)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1314_unique__quotient__lemma,axiom,
    ! [B2: int,Q4: int,R4: int,Q2: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),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),Q2)),R)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R4))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q4),Q2)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1315_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R: int,B3: 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),Q2)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q4)),R4) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q4)),R4)),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),B2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R4))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q4),Q2)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1316_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R: int,B3: 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),Q2)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q4)),R4) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q4)),R4)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B3))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q2),Q4)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1317_q__pos__lemma,axiom,
    ! [B3: int,Q4: int,R4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q4)),R4)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q4)) ) ) ) ).

% q_pos_lemma
tff(fact_1318_incr__mult__lemma,axiom,
    ! [D3: int,P2: fun(int,bool),K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int] :
            ( pp(aa(int,bool,P2,X3))
           => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D3))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
         => ! [X5: int] :
              ( pp(aa(int,bool,P2,X5))
             => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X5),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1319_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_1320_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_1321_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_1322_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)),W2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W2)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W2)) ).

% int_distrib(3)
tff(fact_1323_int__distrib_I4_J,axiom,
    ! [W2: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W2),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z22)) ).

% int_distrib(4)
tff(fact_1324_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C3: A] : divide_divide(A,divide_divide(A,A3,B2),C3) = divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ).

% divide_divide_eq_left'
tff(fact_1325_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,Z2: A,W2: A] : divide_divide(A,divide_divide(A,X,Y),divide_divide(A,Z2,W2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),W2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ).

% divide_divide_times_eq
tff(fact_1326_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,Z2: A,W2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,X,Y)),divide_divide(A,Z2,W2)) = divide_divide(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),W2)) ) ).

% times_divide_times_eq
tff(fact_1327_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C3,D3)))
        <=> ( ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
              | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
                & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),A3))
                  | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D3)) ) ) )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3)) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_1328_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or1337092689740270186AtMost(A,A3,B2))) ) ) ).

% infinite_Icc
tff(fact_1329_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
         => ( set_or1337092689740270186AtMost(A,A3,B2) = aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_1330_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C3)),divide_divide(A,A3,C3))) ) ) ) ).

% divide_right_mono_neg
tff(fact_1331_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_1332_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_1333_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_1334_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_1335_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,A3,B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_1336_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,A3,C3)),divide_divide(A,B2,C3))) ) ) ) ).

% divide_right_mono
tff(fact_1337_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,A3,B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).

% divide_le_0_iff
tff(fact_1338_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,A3,C3)),divide_divide(A,B2,C3))) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1339_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,A3,C3)),divide_divide(A,B2,C3))) ) ) ) ).

% divide_strict_right_mono
tff(fact_1340_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,A3,B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_1341_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,A3,C3)),divide_divide(A,B2,C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
            & ( C3 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_1342_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,A3,B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).

% divide_less_0_iff
tff(fact_1343_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_pos_pos
tff(fact_1344_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_pos_neg
tff(fact_1345_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_neg_pos
tff(fact_1346_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_neg_neg
tff(fact_1347_all__nat__less,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ! [M6: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M6),N))
         => pp(aa(nat,bool,P2,M6)) )
    <=> ! [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P2,X4)) ) ) ).

% all_nat_less
tff(fact_1348_ex__nat__less,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ? [M6: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M6),N))
          & pp(aa(nat,bool,P2,M6)) )
    <=> ? [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P2,X4)) ) ) ).

% ex_nat_less
tff(fact_1349_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z2: A,X: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z2 != zero_zero(A) )
           => ( ( divide_divide(A,X,Y) = 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_1350_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,A3: A] :
          ( ( divide_divide(A,B2,C3) = A3 )
        <=> ( ( ( C3 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) ) )
            & ( ( C3 = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq
tff(fact_1351_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( A3 = divide_divide(A,B2,C3) )
        <=> ( ( ( C3 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = B2 ) )
            & ( ( C3 = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq
tff(fact_1352_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,B2: A,A3: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) )
           => ( divide_divide(A,B2,C3) = A3 ) ) ) ) ).

% divide_eq_imp
tff(fact_1353_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = B2 )
           => ( A3 = divide_divide(A,B2,C3) ) ) ) ) ).

% eq_divide_imp
tff(fact_1354_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,B2: A,A3: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( divide_divide(A,B2,C3) = A3 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1355_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( A3 = divide_divide(A,B2,C3) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1356_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( divide_divide(A,A3,B2) = one_one(A) )
          <=> ( A3 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_1357_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] : divide_divide(A,A3,aa(num,A,numeral_numeral(A),one2)) = A3 ) ).

% divide_numeral_1
tff(fact_1358_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,A3,B2))) ) ) ) ).

% div_positive
tff(fact_1359_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( divide_divide(A,A3,B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1360_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_nonpos_pos
tff(fact_1361_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_nonpos_neg
tff(fact_1362_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_nonneg_pos
tff(fact_1363_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_nonneg_neg
tff(fact_1364_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,A3,C3)),divide_divide(A,B2,C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% divide_le_cancel
tff(fact_1365_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W2: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W2),Z2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Z2)),divide_divide(A,Y,W2))) ) ) ) ) ) ).

% frac_less2
tff(fact_1366_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W2: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Z2)),divide_divide(A,Y,W2))) ) ) ) ) ) ).

% frac_less
tff(fact_1367_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,W2: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Z2)),divide_divide(A,Y,W2))) ) ) ) ) ) ).

% frac_le
tff(fact_1368_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
         => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A3,B2),C3) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1369_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C3)),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% divide_less_eq
tff(fact_1370_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq
tff(fact_1371_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C3)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ).

% neg_divide_less_eq
tff(fact_1372_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% neg_less_divide_eq
tff(fact_1373_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C3)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% pos_divide_less_eq
tff(fact_1374_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ).

% pos_less_divide_eq
tff(fact_1375_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),Z2)) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1376_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),divide_divide(A,X,Y))) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1377_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,C3,A3)),divide_divide(A,C3,B2))) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1378_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,C3,A3)),divide_divide(A,C3,B2))) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1379_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),divide_divide(A,B2,A3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% less_divide_eq_1
tff(fact_1380_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,A3)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_1381_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,W2: num] :
          ( ( divide_divide(A,B2,C3) = aa(num,A,numeral_numeral(A),W2) )
        <=> ( ( ( C3 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3) ) )
            & ( ( C3 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1382_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C3: A] :
          ( ( aa(num,A,numeral_numeral(A),W2) = divide_divide(A,B2,C3) )
        <=> ( ( ( C3 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3) = B2 ) )
            & ( ( C3 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1383_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,A3: A,B2: A] :
          ( ( ( Z2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,Z2)),B2) = B2 ) )
          & ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,Z2)),B2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2)),Z2) ) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1384_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,A3: A,B2: A] :
          ( ( ( Z2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),divide_divide(A,B2,Z2)) = A3 ) )
          & ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),divide_divide(A,B2,Z2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z2)),B2),Z2) ) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1385_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),divide_divide(A,X,Y)),divide_divide(A,W2,Z2)) = 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_1386_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),divide_divide(A,X,Y)),Z2) = 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_1387_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),divide_divide(A,X,Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),Y) ) ) ) ).

% add_num_frac
tff(fact_1388_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),divide_divide(A,Y,Z2)) = 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_1389_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),divide_divide(A,X,Z2)),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)),Z2) ) ) ) ).

% divide_add_eq_iff
tff(fact_1390_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,A3: A,B2: A] :
          ( ( ( Z2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),divide_divide(A,B2,Z2)) = A3 ) )
          & ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),divide_divide(A,B2,Z2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z2)),B2),Z2) ) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_1391_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,aa(A,fun(A,A),minus_minus(A),divide_divide(A,X,Y)),divide_divide(A,W2,Z2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),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_1392_diff__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),minus_minus(A),X),divide_divide(A,Y,Z2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),Y),Z2) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1393_divide__diff__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),minus_minus(A),divide_divide(A,X,Z2)),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)),Z2) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1394_card__Un__le,axiom,
    ! [A: $tType,A6: set(A),B6: set(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B6))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(A),nat,finite_card(A),B6)))) ).

% card_Un_le
tff(fact_1395_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_1396_nat__mult__div__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)) = divide_divide(nat,M2,N) ) ) ).

% nat_mult_div_cancel1
tff(fact_1397_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( set_or1337092689740270186AtMost(nat,M2,N) = aa(set(nat),set(nat),insert(nat,M2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_1398_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
     => ( set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_1399_atLeastAtMost__insertL,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( aa(set(nat),set(nat),insert(nat,M2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N)) = set_or1337092689740270186AtMost(nat,M2,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_1400_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N6: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N6),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite2(nat),N6)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_1401_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C3)),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% divide_le_eq
tff(fact_1402_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq
tff(fact_1403_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,C3,A3)),divide_divide(A,C3,B2))) ) ) ) ) ).

% divide_left_mono
tff(fact_1404_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C3)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ).

% neg_divide_le_eq
tff(fact_1405_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% neg_le_divide_eq
tff(fact_1406_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C3)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% pos_divide_le_eq
tff(fact_1407_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ).

% pos_le_divide_eq
tff(fact_1408_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),Z2)) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1409_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),divide_divide(A,X,Y))) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1410_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,C3,A3)),divide_divide(A,C3,B2))) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1411_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,A3)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_1412_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% le_divide_eq_1
tff(fact_1413_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C3)),aa(num,A,numeral_numeral(A),W2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1414_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W2)),divide_divide(A,B2,C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1415_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) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),divide_divide(A,W2,Z2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),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_1416_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) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),divide_divide(A,W2,Z2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),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_1417_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,N: nat,M2: nat] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) = divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).

% power_diff
tff(fact_1418_div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
       => ( divide_divide(nat,M2,N) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N)) ) ) ) ).

% div_geq
tff(fact_1419_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,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,real,aa(real,fun(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_1420_L2__set__mult__ineq__lemma,axiom,
    ! [A3: real,C3: real,B2: real,D3: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),times_times(real),A3),C3))),aa(real,real,aa(real,fun(real,real),times_times(real),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,aa(real,fun(nat,real),power_power(real),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(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,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),C3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% L2_set_mult_ineq_lemma
tff(fact_1421_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C3)),aa(num,A,numeral_numeral(A),W2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1422_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),divide_divide(A,B2,C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1423_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ) ).

% half_gt_zero
tff(fact_1424_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% half_gt_zero_iff
tff(fact_1425_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V3: A,R: A,S2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R),S2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),R),aa(A,A,aa(A,fun(A,A),minus_minus(A),V3),U)),S2))),V3)) ) ) ) ) ).

% scaling_mono
tff(fact_1426_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E3))
         => ~ ! [N3: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N3)))),E3)) ) ) ).

% nat_approx_posE
tff(fact_1427_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: nat,M2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
         => ( ( N != zero_zero(nat) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),M2))),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),N)))) ) ) ) ).

% inverse_of_nat_le
tff(fact_1428_triangle__def,axiom,
    ! [N: nat] : nat_triangle(N) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% triangle_def
tff(fact_1429_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,X: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ) ) ) ).

% arith_geo_mean
tff(fact_1430_double__not__eq__Suc__double,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% double_not_eq_Suc_double
tff(fact_1431_Suc__double__not__eq__double,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2)) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% Suc_double_not_eq_double
tff(fact_1432_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_1433_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_1434_div2__Suc__Suc,axiom,
    ! [M2: nat] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,M2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,divide_divide(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% div2_Suc_Suc
tff(fact_1435_insert_H__correct,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( 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),insert(nat,X),bot_bot(set(nat))))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat)))) ) ) ).

% insert'_correct
tff(fact_1436_div__mult__self__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N),N) = M2 ) ) ).

% div_mult_self_is_m
tff(fact_1437_div__mult__self1__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M2),N) = M2 ) ) ).

% div_mult_self1_is_m
tff(fact_1438_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A,C3: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A3,B2)) ) ) ) ).

% div_mult_self1
tff(fact_1439_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A,C3: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A3,B2)) ) ) ) ).

% div_mult_self2
tff(fact_1440_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A3,B2)) ) ) ) ).

% div_mult_self3
tff(fact_1441_zdiv__numeral__Bit0,axiom,
    ! [V3: num,W2: num] : divide_divide(int,aa(num,int,numeral_numeral(int),bit0(V3)),aa(num,int,numeral_numeral(int),bit0(W2))) = divide_divide(int,aa(num,int,numeral_numeral(int),V3),aa(num,int,numeral_numeral(int),W2)) ).

% zdiv_numeral_Bit0
tff(fact_1442_real__divide__square__eq,axiom,
    ! [R: real,A3: real] : divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),R),A3),aa(real,real,aa(real,fun(real,real),times_times(real),R),R)) = divide_divide(real,A3,R) ).

% real_divide_square_eq
tff(fact_1443_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( ( C3 = zero_zero(A) )
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = zero_zero(A) ) )
          & ( ( C3 != zero_zero(A) )
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A3,B2) ) ) ) ) ).

% div_mult_mult1_if
tff(fact_1444_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,A3,B2) ) ) ) ).

% div_mult_mult2
tff(fact_1445_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A3: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A3,B2) ) ) ) ).

% div_mult_mult1
tff(fact_1446_div__by__Suc__0,axiom,
    ! [M2: nat] : divide_divide(nat,M2,aa(nat,nat,suc,zero_zero(nat))) = M2 ).

% div_by_Suc_0
tff(fact_1447_div__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( divide_divide(nat,M2,N) = zero_zero(nat) ) ) ).

% div_less
tff(fact_1448_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A3,B2)) ) ) ) ).

% div_mult_self4
tff(fact_1449_zdiv__zmult2__eq,axiom,
    ! [C3: int,A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C3))
     => ( divide_divide(int,A3,aa(int,int,aa(int,fun(int,int),times_times(int),B2),C3)) = divide_divide(int,divide_divide(int,A3,B2),C3) ) ) ).

% zdiv_zmult2_eq
tff(fact_1450_div__neg__pos__less0,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,A3,B2)),zero_zero(int))) ) ) ).

% div_neg_pos_less0
tff(fact_1451_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,A3,B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1452_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,A3,B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int))) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1453_zdiv__int,axiom,
    ! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,A3,B2)) = divide_divide(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zdiv_int
tff(fact_1454_atLeastAtMostPlus1__int__conv,axiom,
    ! [M2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M2),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)))
     => ( set_or1337092689740270186AtMost(int,M2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)) = aa(set(int),set(int),insert(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)),set_or1337092689740270186AtMost(int,M2,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_1455_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),divide_divide(int,A3,B2)))
      <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A3))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1456_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),divide_divide(int,A3,B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1457_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),divide_divide(int,A3,B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int))) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1458_pos__imp__zdiv__pos__iff,axiom,
    ! [K2: int,I2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),divide_divide(int,I2,K2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),I2)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1459_div__nonpos__pos__le0,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A3,B2)),zero_zero(int))) ) ) ).

% div_nonpos_pos_le0
tff(fact_1460_div__nonneg__neg__le0,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A3,B2)),zero_zero(int))) ) ) ).

% div_nonneg_neg_le0
tff(fact_1461_div__positive__int,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),divide_divide(int,K2,L))) ) ) ).

% div_positive_int
tff(fact_1462_div__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),divide_divide(int,K2,L)))
    <=> ( ( K2 = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ) ).

% div_int_pos_iff
tff(fact_1463_zdiv__mono2__neg,axiom,
    ! [A3: int,B3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A3,B3)),divide_divide(int,A3,B2))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1464_zdiv__mono1__neg,axiom,
    ! [A3: int,A4: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),A4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A4,B2)),divide_divide(int,A3,B2))) ) ) ).

% zdiv_mono1_neg
tff(fact_1465_zdiv__eq__0__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( divide_divide(int,I2,K2) = zero_zero(int) )
    <=> ( ( K2 = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K2)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),I2)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1466_zdiv__mono2,axiom,
    ! [A3: int,B3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A3,B2)),divide_divide(int,A3,B3))) ) ) ) ).

% zdiv_mono2
tff(fact_1467_zdiv__mono1,axiom,
    ! [A3: int,A4: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),A4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A3,B2)),divide_divide(int,A4,B2))) ) ) ).

% zdiv_mono1
tff(fact_1468_int__div__less__self,axiom,
    ! [X: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),K2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,X,K2)),X)) ) ) ).

% int_div_less_self
tff(fact_1469_periodic__finite__ex,axiom,
    ! [D3: int,P2: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => ( ! [X3: int,K: int] :
            ( pp(aa(int,bool,P2,X3))
          <=> pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3)))) )
       => ( ? [X_13: int] : pp(aa(int,bool,P2,X_13))
        <=> ? [X4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D3)))
              & pp(aa(int,bool,P2,X4)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_1470_split__zdiv,axiom,
    ! [P2: fun(int,bool),N: int,K2: int] :
      ( pp(aa(int,bool,P2,divide_divide(int,N,K2)))
    <=> ( ( ( K2 = zero_zero(int) )
         => pp(aa(int,bool,P2,zero_zero(int))) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
         => ! [I: int,J: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),K2))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),J) ) )
             => pp(aa(int,bool,P2,I)) ) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
         => ! [I: int,J: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),J))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),zero_zero(int)))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),J) ) )
             => pp(aa(int,bool,P2,I)) ) ) ) ) ).

% split_zdiv
tff(fact_1471_int__div__neg__eq,axiom,
    ! [A3: int,B2: int,Q2: int,R: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R))
         => ( divide_divide(int,A3,B2) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1472_int__div__pos__eq,axiom,
    ! [A3: int,B2: int,Q2: int,R: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),B2))
         => ( divide_divide(int,A3,B2) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1473_cpmi,axiom,
    ! [D5: int,P2: fun(int,bool),P3: fun(int,bool),B6: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),Z4))
           => ( pp(aa(int,bool,P2,X3))
            <=> pp(aa(int,bool,P3,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D5)))
                 => ! [Xb3: int] :
                      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B6))
                     => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa) ) ) )
             => ( pp(aa(int,bool,P2,X3))
               => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5))) ) )
         => ( ! [X3: int,K: int] :
                ( pp(aa(int,bool,P3,X3))
              <=> pp(aa(int,bool,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D5)))) )
           => ( ? [X_13: int] : pp(aa(int,bool,P2,X_13))
            <=> ( ? [X4: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
                    & pp(aa(int,bool,P3,X4)) )
                | ? [X4: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
                    & ? [Xa3: int] :
                        ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),B6))
                        & pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa3),X4))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_1474_cppi,axiom,
    ! [D5: int,P2: fun(int,bool),P3: fun(int,bool),A6: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D5))
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),X3))
           => ( pp(aa(int,bool,P2,X3))
            <=> pp(aa(int,bool,P3,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),set_or1337092689740270186AtMost(int,one_one(int),D5)))
                 => ! [Xb3: int] :
                      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A6))
                     => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa) ) ) )
             => ( pp(aa(int,bool,P2,X3))
               => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5))) ) )
         => ( ! [X3: int,K: int] :
                ( pp(aa(int,bool,P3,X3))
              <=> pp(aa(int,bool,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D5)))) )
           => ( ? [X_13: int] : pp(aa(int,bool,P2,X_13))
            <=> ( ? [X4: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
                    & pp(aa(int,bool,P3,X4)) )
                | ? [X4: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D5)))
                    & ? [Xa3: int] :
                        ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),A6))
                        & pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa3),X4))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_1475_int__power__div__base,axiom,
    ! [M2: nat,K2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
       => ( divide_divide(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),K2),M2),K2) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1476_div__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K2))
       => ( divide_divide(int,K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),L),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1477_div__le__dividend,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,M2,N)),M2)) ).

% div_le_dividend
tff(fact_1478_div__le__mono,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,M2,K2)),divide_divide(nat,N,K2))) ) ).

% div_le_mono
tff(fact_1479_div__mult2__eq,axiom,
    ! [M2: nat,N: nat,Q2: nat] : divide_divide(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) = divide_divide(nat,divide_divide(nat,M2,N),Q2) ).

% div_mult2_eq
tff(fact_1480_pos__zdiv__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = divide_divide(int,B2,A3) ) ) ).

% pos_zdiv_mult_2
tff(fact_1481_neg__zdiv__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
     => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = divide_divide(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int)),A3) ) ) ).

% neg_zdiv_mult_2
tff(fact_1482_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M2: nat,N: nat] : divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N))) = divide_divide(A,divide_divide(A,A3,aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% div_mult2_eq'
tff(fact_1483_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide(nat,M2,N) = zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
        | ( N = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1484_Suc__div__le__mono,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,M2,N)),divide_divide(nat,aa(nat,nat,suc,M2),N))) ).

% Suc_div_le_mono
tff(fact_1485_less__mult__imp__div__less,axiom,
    ! [M2: nat,I2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,M2,N)),I2)) ) ).

% less_mult_imp_div_less
tff(fact_1486_times__div__less__eq__dividend,axiom,
    ! [N: nat,M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),divide_divide(nat,M2,N))),M2)) ).

% times_div_less_eq_dividend
tff(fact_1487_div__times__less__eq__dividend,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M2,N)),N)),M2)) ).

% div_times_less_eq_dividend
tff(fact_1488_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,K2: num,L: num] : divide_divide(A,divide_divide(A,A3,aa(num,A,numeral_numeral(A),K2)),aa(num,A,numeral_numeral(A),L)) = divide_divide(A,A3,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),K2),L))) ) ).

% div_mult2_numeral_eq
tff(fact_1489_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1490_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1491_div__le__mono2,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,K2,N)),divide_divide(nat,K2,M2))) ) ) ).

% div_le_mono2
tff(fact_1492_div__greater__zero__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),divide_divide(nat,M2,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% div_greater_zero_iff
tff(fact_1493_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,M2,Q2)),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2))) ) ) ).

% div_less_iff_less_mult
tff(fact_1494_div__less__dividend,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,M2,N)),M2)) ) ) ).

% div_less_dividend
tff(fact_1495_div__eq__dividend__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( ( divide_divide(nat,M2,N) = M2 )
      <=> ( N = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1496_div__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
          | ( N = zero_zero(nat) ) )
       => ( divide_divide(nat,M2,N) = zero_zero(nat) ) )
      & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
            | ( N = zero_zero(nat) ) )
       => ( divide_divide(nat,M2,N) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N)) ) ) ) ).

% div_if
tff(fact_1497_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q2))))
       => ( divide_divide(nat,M2,N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_1498_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),divide_divide(nat,N,Q2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),Q2)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1499_dividend__less__times__div,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),divide_divide(nat,M2,N))))) ) ).

% dividend_less_times_div
tff(fact_1500_dividend__less__div__times,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M2,N)),N)))) ) ).

% dividend_less_div_times
tff(fact_1501_split__div,axiom,
    ! [P2: fun(nat,bool),M2: nat,N: nat] :
      ( pp(aa(nat,bool,P2,divide_divide(nat,M2,N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P2,zero_zero(nat))) )
        & ( ( N != zero_zero(nat) )
         => ! [I: nat,J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),N))
             => ( ( M2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I)),J) )
               => pp(aa(nat,bool,P2,I)) ) ) ) ) ) ).

% split_div
tff(fact_1502_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,N: nat,M2: nat] :
          ( ( A3 != zero_zero(A) )
         => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
             => ( divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) ) )
            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
             => ( divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) = divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ) ) ).

% power_diff_power_eq
tff(fact_1503_le__div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
       => ( divide_divide(nat,M2,N) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N)) ) ) ) ).

% le_div_geq
tff(fact_1504_split__div_H,axiom,
    ! [P2: fun(nat,bool),M2: nat,N: nat] :
      ( pp(aa(nat,bool,P2,divide_divide(nat,M2,N)))
    <=> ( ( ( N = zero_zero(nat) )
          & pp(aa(nat,bool,P2,zero_zero(nat))) )
        | ? [Q5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)),M2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q5))))
            & pp(aa(nat,bool,P2,Q5)) ) ) ) ).

% split_div'
tff(fact_1505_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% div_2_gt_zero
tff(fact_1506_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),divide_divide(nat,aa(nat,nat,suc,N),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_1507_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),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),N)),X)),one_one(real))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),one_one(real))),N))) ) ).

% linear_plus_1_le_power
tff(fact_1508_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList2: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),X))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = none(nat) ) ) ) ) ).

% succ_list_to_short
tff(fact_1509_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList2: list(vEBT_VEBT),Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = none(nat) ) ) ) ) ).

% pred_list_to_short
tff(fact_1510_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se5668285175392031749et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_1511_case4_I11_J,axiom,
    ( ( mi != ma )
   => ! [I4: nat] :
        ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m)))
       => ( ( ( vEBT_VEBT_high(ma,na) = I4 )
           => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList2),I4)),vEBT_VEBT_low(ma,na))) )
          & ! [X5: nat] :
              ( ( ( vEBT_VEBT_high(X5,na) = I4 )
                & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList2),I4)),vEBT_VEBT_low(X5,na))) )
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),mi),X5))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),ma)) ) ) ) ) ) ).

% case4(11)
tff(fact_1512_enat__ord__number_I1_J,axiom,
    ! [M2: num,N: num] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M2)),aa(num,extended_enat,numeral_numeral(extended_enat),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),M2)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% enat_ord_number(1)
tff(fact_1513_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).

% le_sup_iff
tff(fact_1514_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).

% sup.bounded_iff
tff(fact_1515_bit__split__inv,axiom,
    ! [X: nat,D3: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(X,D3),vEBT_VEBT_low(X,D3),D3) = X ).

% bit_split_inv
tff(fact_1516_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa(set(int),bool,finite_finite2(int),set_or1337092689740270186AtMost(int,L,U))) ).

% finite_atLeastAtMost_int
tff(fact_1517_high__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_high(X,N) = divide_divide(nat,X,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% high_def
tff(fact_1518_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Ma,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2))) ) ).

% high_bound_aux
tff(fact_1519_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
     => ( vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),X),N) = Y ) ) ).

% high_inv
tff(fact_1520_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
     => ( vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),X),N) = X ) ) ).

% low_inv
tff(fact_1521_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2)) ) ) ) ).

% le_inf_iff
tff(fact_1522_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% inf.bounded_iff
tff(fact_1523_both__member__options__ding,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary),N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))
         => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(Info2,Deg,TreeList2,Summary)),X)) ) ) ) ).

% both_member_options_ding
tff(fact_1524_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),Deg))
     => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary)),X))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_1525_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary)),X))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
        & ( ( X = Mi )
          | ( X = Ma )
          | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
            & pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ).

% member_inv
tff(fact_1526_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),Deg))
       => ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))
         => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary)),X)) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_1527_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2))))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_low(X,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_1528_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2))))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2))) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_1529_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).

% inf_sup_ord(2)
tff(fact_1530_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).

% inf_sup_ord(1)
tff(fact_1531_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).

% inf_le1
tff(fact_1532_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).

% inf_le2
tff(fact_1533_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2)) ) ) ) ).

% le_infE
tff(fact_1534_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2))) ) ) ) ).

% le_infI
tff(fact_1535_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C3),D3))) ) ) ) ).

% inf_mono
tff(fact_1536_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% le_infI1
tff(fact_1537_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).

% le_infI2
tff(fact_1538_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).

% inf.orderE
tff(fact_1539_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% inf.orderI
tff(fact_1540_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F3: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X3: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),Y3)),X3))
         => ( ! [X3: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),Y3)),Y3))
           => ( ! [X3: A,Y3: A,Z: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F3,Y3),Z))) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),F3,X),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_1541_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% le_iff_inf
tff(fact_1542_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb1
tff(fact_1543_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_1544_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% inf_absorb1
tff(fact_1545_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_1546_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% inf.boundedE
tff(fact_1547_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3))) ) ) ) ).

% inf.boundedI
tff(fact_1548_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))) ) ) ) ).

% inf_greatest
tff(fact_1549_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).

% inf.order_iff
tff(fact_1550_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),A3)) ) ).

% inf.cobounded1
tff(fact_1551_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2)) ) ).

% inf.cobounded2
tff(fact_1552_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = A3 ) ) ) ).

% inf.absorb_iff1
tff(fact_1553_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).

% inf.absorb_iff2
tff(fact_1554_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C3)) ) ) ).

% inf.coboundedI1
tff(fact_1555_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C3)) ) ) ).

% inf.coboundedI2
tff(fact_1556_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.coboundedI2
tff(fact_1557_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% sup.coboundedI1
tff(fact_1558_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb_iff2
tff(fact_1559_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb_iff1
tff(fact_1560_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ).

% sup.cobounded2
tff(fact_1561_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ).

% sup.cobounded1
tff(fact_1562_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).

% sup.order_iff
tff(fact_1563_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A3)) ) ) ) ).

% sup.boundedI
tff(fact_1564_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).

% sup.boundedE
tff(fact_1565_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_1566_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = X ) ) ) ).

% sup_absorb1
tff(fact_1567_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_1568_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).

% sup.absorb1
tff(fact_1569_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F3: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X3: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F3,X3),Y3)))
         => ( ! [X3: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),aa(A,A,aa(A,fun(A,A),F3,X3),Y3)))
           => ( ! [X3: A,Y3: A,Z: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,Y3),Z)),X3)) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),F3,X),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_1570_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% sup.orderI
tff(fact_1571_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).

% sup.orderE
tff(fact_1572_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_1573_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X)) ) ) ) ).

% sup_least
tff(fact_1574_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C3),D3))) ) ) ) ).

% sup_mono
tff(fact_1575_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A3: A,D3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C3),D3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ) ).

% sup.mono
tff(fact_1576_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% le_supI2
tff(fact_1577_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).

% le_supI1
tff(fact_1578_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% sup_ge2
tff(fact_1579_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% sup_ge1
tff(fact_1580_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X)) ) ) ) ).

% le_supI
tff(fact_1581_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X)) ) ) ) ).

% le_supE
tff(fact_1582_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% inf_sup_ord(3)
tff(fact_1583_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% inf_sup_ord(4)
tff(fact_1584_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
         => vEBT_invar_vebt(X3,N) )
     => ( vEBT_invar_vebt(Summary,M2)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
         => ( ( M2 = N )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
             => ( ! [I3: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2)))
                   => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_13))
                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_12)) ) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),Ma))
                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2)))
                             => ( ( ( vEBT_VEBT_high(Ma,N) = I3 )
                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma,N))) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,N) = I3 )
                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X3,N))) )
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X3))
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Ma)) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_1585_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
         => vEBT_invar_vebt(X3,N) )
     => ( vEBT_invar_vebt(Summary,M2)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
         => ( ( M2 = aa(nat,nat,suc,N) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
             => ( ! [I3: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2)))
                   => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_13))
                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),I3)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                       => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_12)) ) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),Ma))
                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2)))
                             => ( ( ( vEBT_VEBT_high(Ma,N) = I3 )
                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma,N))) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,N) = I3 )
                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X3,N))) )
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X3))
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Ma)) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_1586_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( vEBT_invar_vebt(A12,A23)
     => ( ( ? [A5: bool,B4: bool] : A12 = vEBT_Leaf(A5,B4)
         => ( A23 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList: list(vEBT_VEBT),N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
              ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2) )
             => ( ( A23 = Deg2 )
               => ( ! [X5: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                     => vEBT_invar_vebt(X5,N3) )
                 => ( vEBT_invar_vebt(Summary2,M)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
                     => ( ( M = N3 )
                       => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),M) )
                         => ( ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),X_1))
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                                 => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X5),X_1)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list(vEBT_VEBT),N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
                ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2) )
               => ( ( A23 = Deg2 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => vEBT_invar_vebt(X5,N3) )
                   => ( vEBT_invar_vebt(Summary2,M)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
                       => ( ( M = aa(nat,nat,suc,N3) )
                         => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),M) )
                           => ( ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),X_1))
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                                   => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X5),X_1)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list(vEBT_VEBT),N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg2,TreeList,Summary2) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                         => vEBT_invar_vebt(X5,N3) )
                     => ( vEBT_invar_vebt(Summary2,M)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
                         => ( ( M = N3 )
                           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),M) )
                             => ( ! [I4: nat] :
                                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
                                   => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),X_13))
                                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X5: vEBT_VEBT] :
                                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                                       => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X5),X_1)) ) )
                                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi2),Ma2))
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
                                             => ( ( ( vEBT_VEBT_high(Ma2,N3) = I4 )
                                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(Ma2,N3))) )
                                                & ! [X5: nat] :
                                                    ( ( ( vEBT_VEBT_high(X5,N3) = I4 )
                                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(X5,N3))) )
                                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),X5))
                                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list(vEBT_VEBT),N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg2,TreeList,Summary2) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X5: vEBT_VEBT] :
                            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                           => vEBT_invar_vebt(X5,N3) )
                       => ( vEBT_invar_vebt(Summary2,M)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
                           => ( ( M = aa(nat,nat,suc,N3) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),M) )
                               => ( ! [I4: nat] :
                                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
                                     => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),X_13))
                                      <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),I4)) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X5: vEBT_VEBT] :
                                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                                         => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X5),X_1)) ) )
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi2),Ma2))
                                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
                                               => ( ( ( vEBT_VEBT_high(Ma2,N3) = I4 )
                                                   => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(Ma2,N3))) )
                                                  & ! [X5: nat] :
                                                      ( ( ( vEBT_VEBT_high(X5,N3) = I4 )
                                                        & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(X5,N3))) )
                                                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),X5))
                                                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_1587_invar__vebt_Osimps,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( vEBT_invar_vebt(A12,A23)
    <=> ( ( ? [A7: bool,B5: bool] : A12 = vEBT_Leaf(A7,B5)
          & ( A23 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList4: list(vEBT_VEBT),N5: nat,Summary4: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),A23,TreeList4,Summary4) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4)))
               => vEBT_invar_vebt(X4,N5) )
            & vEBT_invar_vebt(Summary4,N5)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList4) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),N5) )
            & ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary4),X_13))
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4)))
               => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_13)) ) )
        | ? [TreeList4: list(vEBT_VEBT),N5: nat,Summary4: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),A23,TreeList4,Summary4) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4)))
               => vEBT_invar_vebt(X4,N5) )
            & vEBT_invar_vebt(Summary4,aa(nat,nat,suc,N5))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList4) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N5)) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),aa(nat,nat,suc,N5)) )
            & ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary4),X_13))
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4)))
               => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_13)) ) )
        | ? [TreeList4: list(vEBT_VEBT),N5: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A12 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A23,TreeList4,Summary4) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4)))
               => vEBT_invar_vebt(X4,N5) )
            & vEBT_invar_vebt(Summary4,N5)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList4) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),N5) )
            & ! [I: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5)))
               => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I)),X_13))
                <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary4),I)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4)))
                 => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_13)) ) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi3),Ma3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A23)))
            & ( ( Mi3 != Ma3 )
             => ! [I: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5)))
                 => ( ( ( vEBT_VEBT_high(Ma3,N5) = I )
                     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I)),vEBT_VEBT_low(Ma3,N5))) )
                    & ! [X4: nat] :
                        ( ( ( vEBT_VEBT_high(X4,N5) = I )
                          & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I)),vEBT_VEBT_low(X4,N5))) )
                       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi3),X4))
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Ma3)) ) ) ) ) ) )
        | ? [TreeList4: list(vEBT_VEBT),N5: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A12 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A23,TreeList4,Summary4) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4)))
               => vEBT_invar_vebt(X4,N5) )
            & vEBT_invar_vebt(Summary4,aa(nat,nat,suc,N5))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList4) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N5)) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),aa(nat,nat,suc,N5)) )
            & ! [I: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N5))))
               => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I)),X_13))
                <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary4),I)) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4)))
                 => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_13)) ) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi3),Ma3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A23)))
            & ( ( Mi3 != Ma3 )
             => ! [I: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N5))))
                 => ( ( ( vEBT_VEBT_high(Ma3,N5) = I )
                     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I)),vEBT_VEBT_low(Ma3,N5))) )
                    & ! [X4: nat] :
                        ( ( ( vEBT_VEBT_high(X4,N5) = I )
                          & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I)),vEBT_VEBT_low(X4,N5))) )
                       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi3),X4))
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Ma3)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_1588_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),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_1589_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),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_1590_in__children__def,axiom,
    ! [N: nat,TreeList2: list(vEBT_VEBT),X: nat] :
      ( vEBT_V5917875025757280293ildren(N,TreeList2,X)
    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,N))),vEBT_VEBT_low(X,N))) ) ).

% in_children_def
tff(fact_1591_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat,Va3: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),N)
     => ( ( N = aa(nat,nat,suc,aa(nat,nat,suc,Va3)) )
       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),Mi))
         => ( ( Ma != Mi )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Va3,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(Summary)))))),aa(nat,nat,suc,divide_divide(nat,Va3,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ).

% nested_mint
tff(fact_1592_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),Deg)
     => ( ( Mi != Ma )
       => ( aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Summary)) = vEBT_VEBT_high(Ma,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% summaxma
tff(fact_1593_set__encode__insert,axiom,
    ! [A6: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
     => ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),A6))
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),insert(nat,N),A6)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(set(nat),nat,nat_set_encode,A6)) ) ) ) ).

% set_encode_insert
tff(fact_1594_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se2638667681897837118et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_1595_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(Xn,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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList2,H,Newnode) )
                     => ( ~ pp(vEBT_VEBT_minNull(Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xn),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_1596_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L) )
               => ( ~ pp(vEBT_VEBT_minNull(Newnode))
                 => ( ( Newlist = list_update(vEBT_VEBT,TreeList2,H,Newnode) )
                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                     => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_1597_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q2: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),zero_zero(int)))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_1598_length__list__update,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,X: A] : aa(list(A),nat,size_size(list(A)),list_update(A,Xs,I2,X)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_list_update
tff(fact_1599_option_Ocollapse,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) = Option ) ) ).

% option.collapse
tff(fact_1600_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2))
     => ( list_update(A,Xs,I2,X) = Xs ) ) ).

% list_update_beyond
tff(fact_1601_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_1602_nth__list__update__eq,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),I2) = X ) ) ).

% nth_list_update_eq
tff(fact_1603_set__swap,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),set(A),set2(A),list_update(A,list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),J2)),J2,aa(nat,A,nth(A,Xs),I2))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_1604_unique__quotient,axiom,
    ! [A3: int,B2: int,Q2: int,R: int,Q4: int,R4: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R4))
       => ( Q2 = Q4 ) ) ) ).

% unique_quotient
tff(fact_1605_unique__remainder,axiom,
    ! [A3: int,B2: int,Q2: int,R: int,Q4: int,R4: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R4))
       => ( R = R4 ) ) ) ).

% unique_remainder
tff(fact_1606_zip__update,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I2: nat,X: A,Ys: list(B),Y: B] : zip(A,B,list_update(A,Xs,I2,X),list_update(B,Ys,I2,Y)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys),I2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).

% zip_update
tff(fact_1607_option_Osel,axiom,
    ! [A: $tType,X2: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X2)) = X2 ).

% option.sel
tff(fact_1608_option_Oexpand,axiom,
    ! [A: $tType,Option: option(A),Option2: option(A)] :
      ( ( ( Option = none(A) )
      <=> ( Option2 = none(A) ) )
     => ( ( ( Option != none(A) )
         => ( ( Option2 != none(A) )
           => ( aa(option(A),A,the2(A),Option) = aa(option(A),A,the2(A),Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_1609_eucl__rel__int__by0,axiom,
    ! [K2: int] : eucl_rel_int(K2,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K2)) ).

% eucl_rel_int_by0
tff(fact_1610_div__int__unique,axiom,
    ! [K2: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( divide_divide(int,K2,L) = Q2 ) ) ).

% div_int_unique
tff(fact_1611_list__update__code_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I2: nat,Y: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(nat,nat,suc,I2),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),list_update(A,Xs,I2,Y)) ).

% list_update_code(3)
tff(fact_1612_list__update__code_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),zero_zero(nat),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs) ).

% list_update_code(2)
tff(fact_1613_set__encode__eq,axiom,
    ! [A6: set(nat),B6: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
     => ( pp(aa(set(nat),bool,finite_finite2(nat),B6))
       => ( ( aa(set(nat),nat,nat_set_encode,A6) = aa(set(nat),nat,nat_set_encode,B6) )
        <=> ( A6 = B6 ) ) ) ) ).

% set_encode_eq
tff(fact_1614_option_Oexhaust__sel,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) ) ) ).

% option.exhaust_sel
tff(fact_1615_eucl__rel__int__dividesI,axiom,
    ! [L: int,K2: int,Q2: int] :
      ( ( L != zero_zero(int) )
     => ( ( K2 = aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L) )
       => eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_1616_set__update__memI,axiom,
    ! [A: $tType,N: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),list_update(A,Xs,N,X)))) ) ).

% set_update_memI
tff(fact_1617_list__update__same__conv,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ( list_update(A,Xs,I2,X) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I2) = X ) ) ) ).

% list_update_same_conv
tff(fact_1618_nth__list__update,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J2: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ( ( I2 = J2 )
         => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),J2) = X ) )
        & ( ( I2 != J2 )
         => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),J2) = aa(nat,A,nth(A,Xs),J2) ) ) ) ) ).

% nth_list_update
tff(fact_1619_set__encode__inf,axiom,
    ! [A6: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),A6))
     => ( aa(set(nat),nat,nat_set_encode,A6) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_1620_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
    <=> ( ( K2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q2)),R) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),L)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),R))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int))) ) )
            & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( Q2 = zero_zero(int) ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_1621_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q2: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B2))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),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))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_1622_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList2: list(vEBT_VEBT),X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mi),Ma))),Deg,list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_1623_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( X != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Ma))),Deg,list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_1624_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList2: list(vEBT_VEBT),Mi: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X),aa(nat,option(nat),some(nat),Mi),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))) ) ) ) ) ).

% pred_less_length_list
tff(fact_1625_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma))
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X),aa(nat,option(nat),some(nat),Mi),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) ) ).

% pred_lesseq_max
tff(fact_1626_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),X))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) ) ).

% succ_greatereq_min
tff(fact_1627_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList2: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),X))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))) ) ) ) ) ).

% succ_less_length_list
tff(fact_1628_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),X))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = if(vEBT_VEBT,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),Mi)),if(nat,fconj(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Mi)),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))))))),Ma)),aa(bool,bool,aa(bool,fun(bool,bool),fimplies,aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Mi))),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma))),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat)),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),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(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),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,list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))),Ma))),Deg,list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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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mmary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),Ma))),Deg,list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),X),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),Summary)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary)) ) ) ) ) ).

% del_in_range
tff(fact_1629_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list(vEBT_VEBT),L: nat] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(Xn,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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                 => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xn),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none(nat)),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,list_update(vEBT_VEBT,TreeList2,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)))))))),Ma))),Deg,list_update(vEBT_VEBT,TreeList2,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L)),vEBT_vebt_delete(Summary,H)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xn),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,list_update(vEBT_VEBT,TreeList2,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L))),H)))),Ma))),Deg,list_update(vEBT_VEBT,TreeList2,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L)),Summary)) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_1630_set__vebt_H__def,axiom,
    ! [T2: vEBT_VEBT] : vEBT_VEBT_set_vebt(T2) = aa(fun(nat,bool),set(nat),collect(nat),vEBT_vebt_member(T2)) ).

% set_vebt'_def
tff(fact_1631_pred__empty,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_pred(T2,X) = none(nat) )
      <=> ( aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aa(vEBT_VEBT,fun(nat,fun(nat,bool)),T2),X)) = bot_bot(set(nat)) ) ) ) ).

% pred_empty
tff(fact_1632_succ__empty,axiom,
    ! [T2: vEBT_VEBT,N: nat,X: nat] :
      ( vEBT_invar_vebt(T2,N)
     => ( ( vEBT_vebt_succ(T2,X) = none(nat) )
      <=> ( aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ab(vEBT_VEBT,fun(nat,fun(nat,bool)),T2),X)) = bot_bot(set(nat)) ) ) ) ).

% succ_empty
tff(fact_1633_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).

% max.bounded_iff
tff(fact_1634_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_1635_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb1
tff(fact_1636_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_1637_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_1638_max__Suc__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M2),N)) ).

% max_Suc_Suc
tff(fact_1639_max__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B2) = zero_zero(nat) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_1640_max__nat_Oleft__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A3) = A3 ).

% max_nat.left_neutral
tff(fact_1641_max__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B2) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_1642_max__nat_Oright__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),zero_zero(nat)) = A3 ).

% max_nat.right_neutral
tff(fact_1643_max__0L,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),N) = N ).

% max_0L
tff(fact_1644_max__0R,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),zero_zero(nat)) = N ).

% max_0R
tff(fact_1645_finite__Collect__le__nat,axiom,
    ! [K2: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ac(nat,fun(nat,bool)),K2)))) ).

% finite_Collect_le_nat
tff(fact_1646_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)) = aa(num,A,numeral_numeral(A),V3) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(1)
tff(fact_1647_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_1648_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_1649_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_1650_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_1651_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_1652_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_1653_card__Collect__le__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ac(nat,fun(nat,bool)),N))) = aa(nat,nat,suc,N) ).

% card_Collect_le_nat
tff(fact_1654_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
               => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none(nat)),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,list_update(vEBT_VEBT,TreeList2,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)))))))),Ma))),Deg,list_update(vEBT_VEBT,TreeList2,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L)),vEBT_vebt_delete(Summary,H)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,list_update(vEBT_VEBT,TreeList2,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L))),H)))),Ma))),Deg,list_update(vEBT_VEBT,TreeList2,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L)),Summary)) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_1655_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list(vEBT_VEBT),Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L) )
               => ( pp(vEBT_VEBT_minNull(Newnode))
                 => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList2,H,Newnode) )
                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(Sn)),none(nat)),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Sn)))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Sn))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_1656_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L) )
               => ( ( Newlist = list_update(vEBT_VEBT,TreeList2,H,Newnode) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                   => ( ( pp(vEBT_VEBT_minNull(Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none(nat)),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)))))))),Ma))),Deg,Newlist,vEBT_vebt_delete(Summary,H)) ) )
                      & ( ~ pp(vEBT_VEBT_minNull(Newnode))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_1657_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = if(vEBT_VEBT,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))))))),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))))))),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),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,list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))),Ma))),Deg,list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_delete(Summary,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))))))),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))))))),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),Ma))),Deg,list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),Summary)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary)) ) ) ) ) ).

% del_x_mia
tff(fact_1658_del__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT),Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(Xn,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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList2,H,Newnode) )
                     => ( pp(vEBT_VEBT_minNull(Newnode))
                       => ( ( Sn = vEBT_vebt_delete(Summary,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xn),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(Sn)),none(nat)),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Sn)))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Sn))))))),Ma))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_1659_del__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( X = Mi )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Ma)) )
     => ( ( Mi != Ma )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
         => ( ( vEBT_VEBT_high(Xn,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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xn,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList2,H,Newnode) )
                     => ( ( pp(vEBT_VEBT_minNull(Newnode))
                         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xn),Ma),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))),none(nat)),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,H)))))))),Ma))),Deg,Newlist,vEBT_vebt_delete(Summary,H)) ) )
                        & ( ~ pp(vEBT_VEBT_minNull(Newnode))
                         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xn),Ma),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_1660_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_1661_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X5: A,Xa: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X5),Xa) = Xa ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X5),Xa) = X5 ) ) ) ) ).

% max_def_raw
tff(fact_1662_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_1663_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_ad(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_1664_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ae(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_1665_finite__M__bounded__by__nat,axiom,
    ! [P2: fun(nat,bool),I2: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_af(fun(nat,bool),fun(nat,fun(nat,bool)),P2),I2)))) ).

% finite_M_bounded_by_nat
tff(fact_1666_finite__less__ub,axiom,
    ! [F3: fun(nat,nat),U: nat] :
      ( ! [N3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),aa(nat,nat,F3,N3)))
     => pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ag(fun(nat,nat),fun(nat,fun(nat,bool)),F3),U)))) ) ).

% finite_less_ub
tff(fact_1667_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A3: A,D3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C3),D3)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ) ).

% max.mono
tff(fact_1668_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).

% max.orderE
tff(fact_1669_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% max.orderI
tff(fact_1670_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A3))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3)) ) ) ) ).

% max.boundedE
tff(fact_1671_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A3)) ) ) ) ).

% max.boundedI
tff(fact_1672_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).

% max.order_iff
tff(fact_1673_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ).

% max.cobounded1
tff(fact_1674_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ).

% max.cobounded2
tff(fact_1675_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z2: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y)) ) ) ) ).

% le_max_iff_disj
tff(fact_1676_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).

% max.absorb_iff1
tff(fact_1677_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_1678_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.coboundedI1
tff(fact_1679_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).

% max.coboundedI2
tff(fact_1680_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_1681_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = X ) ) ) ).

% max_absorb1
tff(fact_1682_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = B2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ) ).

% max_def
tff(fact_1683_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_1684_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_1685_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2)) ) ).

% max_diff_distrib_left
tff(fact_1686_nat__add__max__left,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M2),N)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Q2)) ).

% nat_add_max_left
tff(fact_1687_nat__add__max__right,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),Q2)) ).

% nat_add_max_right
tff(fact_1688_set__vebt__def,axiom,
    ! [T2: vEBT_VEBT] : vEBT_set_vebt(T2) = aa(fun(nat,bool),set(nat),collect(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2)) ).

% set_vebt_def
tff(fact_1689_nat__mult__max__left,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M2),N)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ).

% nat_mult_max_left
tff(fact_1690_nat__mult__max__right,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),Q2)) ).

% nat_mult_max_right
tff(fact_1691_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bit0(N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_code(2)
tff(fact_1692_nat__leq__as__int,axiom,
    ! [X5: nat,Xa: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),Xa))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).

% nat_leq_as_int
tff(fact_1693_power__numeral__even,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z2: A,W2: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),bit0(W2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2))) ) ).

% power_numeral_even
tff(fact_1694_card__less,axiom,
    ! [M5: set(nat),I2: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M5))
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ah(set(nat),fun(nat,fun(nat,bool)),M5),I2))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_1695_card__less__Suc,axiom,
    ! [M5: set(nat),I2: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M5))
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ai(set(nat),fun(nat,fun(nat,bool)),M5),I2)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ah(set(nat),fun(nat,fun(nat,bool)),M5),I2))) ) ) ).

% card_less_Suc
tff(fact_1696_card__less__Suc2,axiom,
    ! [M5: set(nat),I2: nat] :
      ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M5))
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ai(set(nat),fun(nat,fun(nat,bool)),M5),I2))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ah(set(nat),fun(nat,fun(nat,bool)),M5),I2))) ) ) ).

% card_less_Suc2
tff(fact_1697_nat__minus__add__max,axiom,
    ! [N: nat,M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)),M2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),M2) ).

% nat_minus_add_max
tff(fact_1698_finite__lists__length__eq,axiom,
    ! [A: $tType,A6: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_aj(set(A),fun(nat,fun(list(A),bool)),A6),N)))) ) ).

% finite_lists_length_eq
tff(fact_1699_card__lists__length__eq,axiom,
    ! [A: $tType,A6: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_aj(set(A),fun(nat,fun(list(A),bool)),A6),N))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A6)),N) ) ) ).

% card_lists_length_eq
tff(fact_1700_finite__lists__length__le,axiom,
    ! [A: $tType,A6: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_ak(set(A),fun(nat,fun(list(A),bool)),A6),N)))) ) ).

% finite_lists_length_le
tff(fact_1701_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary),X) = if(vEBT_VEBT,fconj(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(bool,bool,fNot,fdisj(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Mi),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma)))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),X,Mi)),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X)),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi),Mi,X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary)) ).

% vebt_insert.simps(5)
tff(fact_1702_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2),X)
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
         => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_1703_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2),X)
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_1704_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( ! [A5: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A5,B4) )
           => ~ ( ( ( Xa2 = zero_zero(nat) )
                 => ( Y = vEBT_Leaf(fTrue,B4) ) )
                & ( ( Xa2 != zero_zero(nat) )
                 => ( ( ( Xa2 = one_one(nat) )
                     => ( Y = vEBT_Leaf(A5,fTrue) ) )
                    & ( ( Xa2 != one_one(nat) )
                     => ( Y = vEBT_Leaf(A5,B4) ) ) ) ) ) )
       => ( ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info,zero_zero(nat),Ts,S) )
             => ( Y != vEBT_Node(Info,zero_zero(nat),Ts,S) ) )
         => ( ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S) )
               => ( Y != vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S) ) )
           => ( ! [V2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary2) )
                 => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),Xa2)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) )
                   => ( Y != if(vEBT_VEBT,fconj(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(bool,bool,fNot,fdisj(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Mi2),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Ma2)))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Xa2,Mi2)),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2)),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary2)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_1705_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary)),X))
    <=> ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
             => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X))
                & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X))
                 => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
                     => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_1706_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V3),TreeList2,Vc),X)
    <=> ( ( X = Mi )
        | ( X = Ma )
        | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)))
           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_1707_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa2)
     => ( ! [A5: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A5,B4) )
           => ( ( ( Xa2 = zero_zero(nat) )
               => pp(A5) )
              & ( ( Xa2 != zero_zero(nat) )
               => ( ( ( Xa2 = one_one(nat) )
                   => pp(B4) )
                  & ( Xa2 = one_one(nat) ) ) ) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw)
         => ~ ! [Uy: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S: vEBT_VEBT] : X = vEBT_Node(Uy,aa(nat,nat,suc,V2),TreeList,S)
               => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                   => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_1708_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_V5719532721284313246member(X,Xa2)
     => ( ! [A5: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A5,B4) )
           => ~ ( ( ( Xa2 = zero_zero(nat) )
                 => pp(A5) )
                & ( ( Xa2 != zero_zero(nat) )
                 => ( ( ( Xa2 = one_one(nat) )
                     => pp(B4) )
                    & ( Xa2 = one_one(nat) ) ) ) ) )
       => ~ ! [Uy: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [S: vEBT_VEBT] : X = vEBT_Node(Uy,aa(nat,nat,suc,V2),TreeList,S)
             => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                   => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_1709_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_V5719532721284313246member(X,Xa2)
      <=> pp(Y) )
     => ( ! [A5: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A5,B4) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero(nat) )
                   => pp(A5) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => pp(B4) )
                      & ( Xa2 = one_one(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw)
           => pp(Y) )
         => ~ ! [Uy: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S: vEBT_VEBT] : X = vEBT_Node(Uy,aa(nat,nat,suc,V2),TreeList,S)
               => ( pp(Y)
                <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_1710_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_VEBT_membermima(X,Xa2)
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va2,Vb2)
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList,Vc2)
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                     => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) )
         => ~ ! [V2: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd)
               => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                     => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_1711_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
     => ( ! [A5: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A5,B4) )
           => ~ ( ( ( Xa2 = zero_zero(nat) )
                 => pp(A5) )
                & ( ( Xa2 != zero_zero(nat) )
                 => ( ( ( Xa2 = one_one(nat) )
                     => pp(B4) )
                    & ( Xa2 = one_one(nat) ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                           => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                               => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_1712_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa2)
     => ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va2,Vb2)
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList,Vc2)
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                       => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) )
             => ~ ! [V2: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd)
                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                       => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_1713_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_VEBT_membermima(X,Xa2)
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => pp(Y) )
       => ( ( ? [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)
           => pp(Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va2,Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList,Vc2)
                 => ( pp(Y)
                  <=> ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) )
             => ~ ! [V2: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd)
                   => ( pp(Y)
                    <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_1714_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
     => ( ! [A5: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A5,B4) )
           => ( ( ( Xa2 = zero_zero(nat) )
               => pp(A5) )
              & ( ( Xa2 != zero_zero(nat) )
               => ( ( ( Xa2 = one_one(nat) )
                   => pp(B4) )
                  & ( Xa2 = one_one(nat) ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw)
         => ( ! [V2: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy,Uz2)
           => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                           => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                               => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                                   => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_1715_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( ! [A5: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A5,B4) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa2 = zero_zero(nat) )
                   => pp(A5) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => pp(B4) )
                      & ( Xa2 = one_one(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw)
           => pp(Y) )
         => ( ( ? [V2: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy,Uz2)
             => pp(Y) )
           => ( ( ? [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => pp(Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)
                   => ( pp(Y)
                    <=> ~ ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                               => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                                  & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                                       => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_1716_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary),X) = aa(nat,option(nat),some(nat),Ma) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X))
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary),X) = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X),aa(nat,option(nat),some(nat),Mi),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_pred(Summary,vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_1717_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary),X) = aa(nat,option(nat),some(nat),Mi) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary),X) = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_succ(Summary,vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_1718_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
          | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X)) )
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary),X) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary) ) )
      & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Mi))
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),X)) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary),X) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary),X) = if(vEBT_VEBT,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary)))))),X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary)))))),X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary)))))),X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary)))))),X),Mi)),if(nat,fconj(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Mi)),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary))))))),Ma)),aa(bool,bool,aa(bool,fun(bool,bool),fimplies,aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Mi))),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Ma))),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary)))))),X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat)),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),X),Mi),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_div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divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary)))))),X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary)))))),X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary)))))),X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary)))))),X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),list_update(vEBT_VEBT,TreeList2,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary)))))),X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary)))))),X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),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(Summary)))))),X),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),Summary)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList2,Summary)) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_1719_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa2) = Y )
     => ( ! [A5: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A5,B4) )
           => ( ( Xa2 = zero_zero(nat) )
             => ( Y != vEBT_Leaf(fFalse,B4) ) ) )
       => ( ! [A5: bool] :
              ( ? [B4: bool] : X = vEBT_Leaf(A5,B4)
             => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != vEBT_Leaf(A5,fFalse) ) ) )
         => ( ! [A5: bool,B4: bool] :
                ( ( X = vEBT_Leaf(A5,B4) )
               => ( ? [N3: nat] : Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,N3))
                 => ( Y != vEBT_Leaf(A5,B4) ) ) )
           => ( ! [Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2) )
                 => ( Y != vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry) )
                   => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr: list(vEBT_VEBT),Sm: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) )
                     => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) )
                       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                                | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2)) )
                             => ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) ) )
                            & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                                  | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2)) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y = 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eral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),Mi2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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(vEBT_vebt_delete(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),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,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),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(vEBT_vebt_delete(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(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) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_1720_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( ! [Uu2: bool,B4: bool] :
            ( ( X = vEBT_Leaf(Uu2,B4) )
           => ( ( Xa2 = zero_zero(nat) )
             => ~ ( ( pp(B4)
                   => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                  & ( ~ pp(B4)
                   => ( Y = none(nat) ) ) ) ) )
       => ( ( ? [Uv2: bool,Uw: bool] : X = vEBT_Leaf(Uv2,Uw)
           => ( ? [N3: nat] : Xa2 = aa(nat,nat,suc,N3)
             => ( 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) ) )
           => ( ( ? [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd)
               => ( Y != none(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != none(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) )
                     => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                           => ( Y = aa(nat,option(nat),some(nat),Mi2) ) )
                          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                           => ( Y = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_1721_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(X,Xa2) = Y )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( ( Xa2 = zero_zero(nat) )
           => ( Y != none(nat) ) ) )
       => ( ! [A5: bool] :
              ( ? [Uw: bool] : X = vEBT_Leaf(A5,Uw)
             => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
               => ~ ( ( pp(A5)
                     => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                    & ( ~ pp(A5)
                     => ( Y = none(nat) ) ) ) ) )
         => ( ! [A5: bool,B4: bool] :
                ( ( X = vEBT_Leaf(A5,B4) )
               => ( ? [Va: nat] : Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,Va))
                 => ~ ( ( pp(B4)
                       => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                      & ( ~ pp(B4)
                       => ( ( pp(A5)
                           => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                          & ( ~ pp(A5)
                           => ( Y = none(nat) ) ) ) ) ) ) )
           => ( ( ? [Uy: nat,Uz2: list(vEBT_VEBT),Va2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va2)
               => ( Y != none(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd,Ve)
                 => ( Y != none(nat) ) )
               => ( ( ? [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != none(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) )
                       => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                             => ( Y = aa(nat,option(nat),some(nat),Ma2) ) )
                            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                             => ( Y = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),Xa2),aa(nat,option(nat),some(nat),Mi2),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_1722_finite__nth__roots,axiom,
    ! [N: nat,C3: complex] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(set(complex),bool,finite_finite2(complex),aa(fun(complex,bool),set(complex),collect(complex),aa(complex,fun(complex,bool),aTP_Lamp_al(nat,fun(complex,fun(complex,bool)),N),C3)))) ) ).

% finite_nth_roots
tff(fact_1723_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_am(nat,fun(A,bool),N)))),N)) ) ) ).

% card_roots_unity
tff(fact_1724_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_am(nat,fun(A,bool),N)))) ) ) ).

% finite_roots_unity
tff(fact_1725_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,B4: bool] :
              ( ( X = vEBT_Leaf(Uu2,B4) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( ( pp(B4)
                     => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                    & ( ~ pp(B4)
                     => ( Y = none(nat) ) ) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,B4)),zero_zero(nat))) ) ) )
         => ( ! [Uv2: bool,Uw: bool] :
                ( ( X = vEBT_Leaf(Uv2,Uw) )
               => ! [N3: nat] :
                    ( ( Xa2 = aa(nat,nat,suc,N3) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uv2,Uw)),aa(nat,nat,suc,N3))) ) ) )
           => ( ! [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2) )
                 => ( ( Y = none(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)),Xa2)) ) )
             => ( ! [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd)),Xa2)) ) )
               => ( ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xa2)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) )
                       => ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                             => ( Y = aa(nat,option(nat),some(nat),Mi2) ) )
                            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                             => ( Y = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_less(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_succ(Summary2,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_1726_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( Y = none(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),zero_zero(nat))) ) ) )
         => ( ! [A5: bool,Uw: bool] :
                ( ( X = vEBT_Leaf(A5,Uw) )
               => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( ( pp(A5)
                       => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                      & ( ~ pp(A5)
                       => ( Y = none(nat) ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,Uw)),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: bool,B4: bool] :
                  ( ( X = vEBT_Leaf(A5,B4) )
                 => ! [Va: nat] :
                      ( ( Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                     => ( ( ( pp(B4)
                           => ( Y = aa(nat,option(nat),some(nat),one_one(nat)) ) )
                          & ( ~ pp(B4)
                           => ( ( pp(A5)
                               => ( Y = aa(nat,option(nat),some(nat),zero_zero(nat)) ) )
                              & ( ~ pp(A5)
                               => ( Y = none(nat) ) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz2: list(vEBT_VEBT),Va2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va2) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va2)),Xa2)) ) )
               => ( ! [V2: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd,Ve) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd,Ve)),Xa2)) ) )
                 => ( ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( Y = none(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xa2)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) )
                         => ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                               => ( Y = aa(nat,option(nat),some(nat),Ma2) ) )
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                               => ( Y = if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(option(nat),fconj(aa(bool,bool,fNot,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat))),vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(option(nat),aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),none(nat)),if(option(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),Xa2),aa(nat,option(nat),some(nat),Mi2),none(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_pred(Summary2,vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),none(nat)) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_1727_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A5: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A5,B4) )
             => ( ( Xa2 = zero_zero(nat) )
               => ( ( Y = vEBT_Leaf(fFalse,B4) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),zero_zero(nat))) ) ) )
         => ( ! [A5: bool,B4: bool] :
                ( ( X = vEBT_Leaf(A5,B4) )
               => ( ( Xa2 = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = vEBT_Leaf(A5,fFalse) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A5: bool,B4: bool] :
                  ( ( X = vEBT_Leaf(A5,B4) )
                 => ! [N3: nat] :
                      ( ( Xa2 = aa(nat,nat,suc,aa(nat,nat,suc,N3)) )
                     => ( ( Y = vEBT_Leaf(A5,B4) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),aa(nat,nat,suc,aa(nat,nat,suc,N3)))) ) ) )
             => ( ! [Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2) )
                   => ( ( Y = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2) )
                     => ~ 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,TreeList,Summary2)),Xa2)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry) )
                     => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst,Smry)),Xa2)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr: list(vEBT_VEBT),Sm: vEBT_VEBT] :
                        ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) )
                       => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm)),Xa2)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) )
                         => ( ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                                  | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2)) )
                               => ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) ) )
                              & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                                    | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2)) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y = if(vEBT_VEBT,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),if(vEBT_VEBT,vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Mi2),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),Mi2)),if(nat,fconj(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Mi2)),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))))))),Ma2)),aa(bool,bool,aa(bool,fun(bool,bool),fimplies,aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Mi2))),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Ma2))),if(nat,aa(option(nat),bool,aa(option(nat),fun(option(nat),bool),fequal(option(nat)),vEBT_vebt_maxt(vEBT_vebt_delete(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),none(nat)),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Mi2),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,num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),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),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,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),Summary2)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_1728_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A5: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A5,B4) )
             => ( ( ( ( Xa2 = zero_zero(nat) )
                   => ( Y = vEBT_Leaf(fTrue,B4) ) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => ( Y = vEBT_Leaf(A5,fTrue) ) )
                      & ( ( Xa2 != one_one(nat) )
                       => ( Y = vEBT_Leaf(A5,B4) ) ) ) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),Xa2)) ) )
         => ( ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,zero_zero(nat),Ts,S) )
               => ( ( Y = vEBT_Node(Info,zero_zero(nat),Ts,S) )
                 => ~ 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(Info,zero_zero(nat),Ts,S)),Xa2)) ) )
           => ( ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S) )
                 => ( ( Y = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S) )
                   => ~ 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(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S)),Xa2)) ) )
             => ( ! [V2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary2) )
                   => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),Xa2)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary2) )
                     => ~ 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,V2)),TreeList,Summary2)),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) )
                     => ( ( Y = if(vEBT_VEBT,fconj(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(bool,bool,fNot,fdisj(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Mi2),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Xa2),Ma2)))),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Xa2,Mi2)),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2)),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),if(vEBT_VEBT,vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary2,vEBT_VEBT_high(if(nat,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2),Mi2,Xa2),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary2)),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_1729_low__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_low(X,N) = modulo_modulo(nat,X,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% low_def
tff(fact_1730_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,zero_zero(A),A3) = zero_zero(A) ) ).

% mod_0
tff(fact_1731_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,zero_zero(A)) = A3 ) ).

% mod_by_0
tff(fact_1732_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,A3) = zero_zero(A) ) ).

% mod_self
tff(fact_1733_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : modulo_modulo(A,zero_zero(A),A3) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_1734_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),B2) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_1735_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),B2) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_1736_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_1737_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : modulo_modulo(A,A3,one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_1738_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A] : divide_divide(A,modulo_modulo(A,A3,B2),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_1739_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B2: A] : divide_divide(A,modulo_modulo(A,A3,B2),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_1740_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self1
tff(fact_1741_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A,C3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self2
tff(fact_1742_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self3
tff(fact_1743_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_mult_self4
tff(fact_1744_mod__by__Suc__0,axiom,
    ! [M2: nat] : modulo_modulo(nat,M2,aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_1745_Suc__mod__mult__self4,axiom,
    ! [N: nat,K2: nat,M2: 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),N),K2)),M2)),N) = modulo_modulo(nat,aa(nat,nat,suc,M2),N) ).

% Suc_mod_mult_self4
tff(fact_1746_Suc__mod__mult__self3,axiom,
    ! [K2: nat,N: nat,M2: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)),M2)),N) = modulo_modulo(nat,aa(nat,nat,suc,M2),N) ).

% Suc_mod_mult_self3
tff(fact_1747_Suc__mod__mult__self2,axiom,
    ! [M2: nat,N: nat,K2: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2))),N) = modulo_modulo(nat,aa(nat,nat,suc,M2),N) ).

% Suc_mod_mult_self2
tff(fact_1748_Suc__mod__mult__self1,axiom,
    ! [M2: nat,K2: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N))),N) = modulo_modulo(nat,aa(nat,nat,suc,M2),N) ).

% Suc_mod_mult_self1
tff(fact_1749_mod2__Suc__Suc,axiom,
    ! [M2: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% mod2_Suc_Suc
tff(fact_1750_Suc__times__numeral__mod__eq,axiom,
    ! [K2: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K2) != 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),K2)),N)),aa(num,nat,numeral_numeral(nat),K2)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1751_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) )
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1752_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) )
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1753_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_1754_add__self__mod__2,axiom,
    ! [M2: nat] : modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),M2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_1755_mod2__gr__0,axiom,
    ! [M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
    <=> ( modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_1756_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C3)),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) ) ).

% mod_mult_eq
tff(fact_1757_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C3: A,A4: A,B2: A,B3: A] :
          ( ( modulo_modulo(A,A3,C3) = modulo_modulo(A,A4,C3) )
         => ( ( modulo_modulo(A,B2,C3) = modulo_modulo(A,B3,C3) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3),C3) ) ) ) ) ).

% mod_mult_cong
tff(fact_1758_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,B2)),C3) ) ).

% mod_mult_mult2
tff(fact_1759_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C3),modulo_modulo(A,A3,B2)) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ).

% mult_mod_right
tff(fact_1760_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,C3)),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) ) ).

% mod_mult_left_eq
tff(fact_1761_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A,C3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) ) ).

% mod_mult_right_eq
tff(fact_1762_mod__Suc__Suc__eq,axiom,
    ! [M2: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,modulo_modulo(nat,M2,N))),N) = modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M2)),N) ).

% mod_Suc_Suc_eq
tff(fact_1763_mod__Suc__eq,axiom,
    ! [M2: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,modulo_modulo(nat,M2,N)),N) = modulo_modulo(nat,aa(nat,nat,suc,M2),N) ).

% mod_Suc_eq
tff(fact_1764_mod__less__eq__dividend,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M2,N)),M2)) ).

% mod_less_eq_dividend
tff(fact_1765_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),modulo_modulo(A,A3,B2)),A3)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1766_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,B2)),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1767_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( modulo_modulo(A,A3,B2) = A3 )
        <=> ( divide_divide(A,A3,B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1768_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,Q2: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(M2)),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(N)),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M2),aa(num,A,numeral_numeral(A),Q2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1769_cong__exp__iff__simps_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),M2),aa(num,A,numeral_numeral(A),one2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),one2)) ) ).

% cong_exp_iff_simps(4)
tff(fact_1770_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C3: A,B2: A] :
          ( ( modulo_modulo(A,A3,C3) = modulo_modulo(A,B2,C3) )
         => ~ ! [D2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D2)) ) ) ).

% mod_eqE
tff(fact_1771_mod__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( aa(nat,nat,suc,modulo_modulo(nat,M2,N)) = N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M2),N) = zero_zero(nat) ) )
      & ( ( aa(nat,nat,suc,modulo_modulo(nat,M2,N)) != N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M2),N) = aa(nat,nat,suc,modulo_modulo(nat,M2,N)) ) ) ) ).

% mod_Suc
tff(fact_1772_mod__induct,axiom,
    ! [P2: fun(nat,bool),N: nat,P: nat,M2: nat] :
      ( pp(aa(nat,bool,P2,N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),P))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),P))
         => ( ! [N3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),P))
               => ( pp(aa(nat,bool,P2,N3))
                 => pp(aa(nat,bool,P2,modulo_modulo(nat,aa(nat,nat,suc,N3),P))) ) )
           => pp(aa(nat,bool,P2,M2)) ) ) ) ) ).

% mod_induct
tff(fact_1773_mod__less__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),modulo_modulo(nat,M2,N)),N)) ) ).

% mod_less_divisor
tff(fact_1774_gcd__nat__induct,axiom,
    ! [P2: fun(nat,fun(nat,bool)),M2: nat,N: nat] :
      ( ! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,M),zero_zero(nat)))
     => ( ! [M: nat,N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,N3),modulo_modulo(nat,M,N3)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,M),N3)) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),P2,M2),N)) ) ) ).

% gcd_nat_induct
tff(fact_1775_mod__Suc__le__divisor,axiom,
    ! [M2: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M2,aa(nat,nat,suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_1776_mod__eq__0D,axiom,
    ! [M2: nat,D3: nat] :
      ( ( modulo_modulo(nat,M2,D3) = zero_zero(nat) )
     => ? [Q3: nat] : M2 = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D3),Q3) ) ).

% mod_eq_0D
tff(fact_1777_mod__geq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
     => ( modulo_modulo(nat,M2,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N) ) ) ).

% mod_geq
tff(fact_1778_le__mod__geq,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( modulo_modulo(nat,M2,N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N) ) ) ).

% le_mod_geq
tff(fact_1779_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y,N) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1780_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( modulo_modulo(A,A3,B2) = A3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1781_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),modulo_modulo(A,A3,B2))) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1782_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(N)),aa(num,A,numeral_numeral(A),bit0(Q2))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1783_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_1784_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(N)),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(6)
tff(fact_1785_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(M2)),aa(num,A,numeral_numeral(A),bit0(Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(8)
tff(fact_1786_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A,C3: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,C3))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),modulo_modulo(A,B2,C3)),C3)) ) ).

% div_mult1_eq
tff(fact_1787_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A3,B2))),modulo_modulo(A,A3,B2))),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) ) ).

% cancel_div_mod_rules(2)
tff(fact_1788_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2)),modulo_modulo(A,A3,B2))),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) ) ).

% cancel_div_mod_rules(1)
tff(fact_1789_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2)),modulo_modulo(A,A3,B2)) ) ).

% mod_div_decomp
tff(fact_1790_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2)),modulo_modulo(A,A3,B2)) = A3 ) ).

% div_mult_mod_eq
tff(fact_1791_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2)) = A3 ) ).

% mod_div_mult_eq
tff(fact_1792_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A3,B2))) = A3 ) ).

% mod_mult_div_eq
tff(fact_1793_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A3,B2))),modulo_modulo(A,A3,B2)) = A3 ) ).

% mult_div_mod_eq
tff(fact_1794_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2)) = modulo_modulo(A,A3,B2) ) ).

% minus_div_mult_eq_mod
tff(fact_1795_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,A3,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_1796_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,A3,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A3,B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_1797_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A3,B2))) = modulo_modulo(A,A3,B2) ) ).

% minus_mult_div_eq_mod
tff(fact_1798_mod__le__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M2,N)),N)) ) ).

% mod_le_divisor
tff(fact_1799_div__less__mono,axiom,
    ! [A6: nat,B6: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A6),B6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( modulo_modulo(nat,A6,N) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B6,N) = zero_zero(nat) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,A6,N)),divide_divide(nat,B6,N))) ) ) ) ) ).

% div_less_mono
tff(fact_1800_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y,N) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
       => ? [Q3: nat] : X = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1801_mod__eq__nat2E,axiom,
    ! [M2: nat,Q2: nat,N: nat] :
      ( ( modulo_modulo(nat,M2,Q2) = modulo_modulo(nat,N,Q2) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ~ ! [S: nat] : N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S)) ) ) ).

% mod_eq_nat2E
tff(fact_1802_mod__eq__nat1E,axiom,
    ! [M2: nat,Q2: nat,N: nat] :
      ( ( modulo_modulo(nat,M2,Q2) = modulo_modulo(nat,N,Q2) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
       => ~ ! [S: nat] : M2 != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S)) ) ) ).

% mod_eq_nat1E
tff(fact_1803_mod__mult2__eq,axiom,
    ! [M2: nat,N: nat,Q2: nat] : modulo_modulo(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),modulo_modulo(nat,divide_divide(nat,M2,N),Q2))),modulo_modulo(nat,M2,N)) ).

% mod_mult2_eq
tff(fact_1804_div__mod__decomp,axiom,
    ! [A6: nat,N: nat] : A6 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,A6,N)),N)),modulo_modulo(nat,A6,N)) ).

% div_mod_decomp
tff(fact_1805_modulo__nat__def,axiom,
    ! [M2: nat,N: nat] : modulo_modulo(nat,M2,N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M2,N)),N)) ).

% modulo_nat_def
tff(fact_1806_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M2: nat,N: nat] : modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N))) = 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),M2)),modulo_modulo(A,divide_divide(A,A3,aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,A,semiring_1_of_nat(A),N)))),modulo_modulo(A,A3,aa(nat,A,semiring_1_of_nat(A),M2))) ) ).

% mod_mult2_eq'
tff(fact_1807_split__mod,axiom,
    ! [P2: fun(nat,bool),M2: nat,N: nat] :
      ( pp(aa(nat,bool,P2,modulo_modulo(nat,M2,N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P2,M2)) )
        & ( ( N != zero_zero(nat) )
         => ! [I: nat,J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),N))
             => ( ( M2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I)),J) )
               => pp(aa(nat,bool,P2,J)) ) ) ) ) ) ).

% split_mod
tff(fact_1808_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
         => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),modulo_modulo(A,divide_divide(A,A3,B2),C3))),modulo_modulo(A,A3,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1809_Suc__times__mod__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M2))
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),M2) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_1810_nth__rotate1,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,rotate1(A,Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,suc,N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate1
tff(fact_1811_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2))
           => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)) = modulo_modulo(A,A3,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1812_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( ( divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_1813_verit__le__mono__div,axiom,
    ! [A6: nat,B6: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A6),B6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,A6,N)),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),modulo_modulo(nat,B6,N)),zero_zero(nat)),one_one(nat),zero_zero(nat)))),divide_divide(nat,B6,N))) ) ) ).

% verit_le_mono_div
tff(fact_1814_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))) = divide_divide(A,A3,B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1815_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1816_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)) = modulo_modulo(A,X,M2) )
              | ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M2)),M2) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_1817_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))))
             => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2) = modulo_modulo(A,A3,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_1818_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,N,divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_1819_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,N,divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_1820_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,bool),set(complex),collect(complex),aTP_Lamp_an(nat,fun(complex,bool),N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_1821_card__nth__roots,axiom,
    ! [C3: complex,N: nat] :
      ( ( C3 != zero_zero(complex) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,bool),set(complex),collect(complex),aa(nat,fun(complex,bool),aTP_Lamp_ao(complex,fun(nat,fun(complex,bool)),C3),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_1822_product__nth,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),N) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),divide_divide(nat,N,aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,N,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_1823_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))))
             => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)))),one_one(A)) = divide_divide(A,A3,B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1824_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A5: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A5,B4) )
             => ( ( Y = vEBT_vebt_insert(vEBT_Leaf(A5,B4),Xa2) )
               => ~ 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(A5,B4)),Xa2)) ) )
         => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
               => ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)),Xa2))
                     => ( Y = vEBT_Node(Info,Deg2,TreeList,Summary2) ) )
                    & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)),Xa2))
                     => ( Y = vEBT_vebt_insert(vEBT_Node(Info,Deg2,TreeList,Summary2),Xa2) ) ) )
                 => ~ 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(Info,Deg2,TreeList,Summary2)),Xa2)) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
tff(fact_1825_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
      <=> pp(Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A5: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A5,B4) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero(nat) )
                     => pp(A5) )
                    & ( ( Xa2 != zero_zero(nat) )
                     => ( ( ( Xa2 = one_one(nat) )
                         => pp(B4) )
                        & ( Xa2 = one_one(nat) ) ) ) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),Xa2)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw) )
               => ( ~ pp(Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw)),Xa2)) ) )
           => ( ! [V2: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy,Uz2) )
                 => ( ~ pp(Y)
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy,Uz2)),Xa2)) ) )
             => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ~ pp(Y)
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xa2)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) )
                     => ( ( pp(Y)
                        <=> ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                                & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                                 => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                                    & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                                     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                                         => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),Xa2)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_1826_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A5: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A5,B4) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),Xa2))
               => ( ( ( Xa2 = zero_zero(nat) )
                   => pp(A5) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => pp(B4) )
                      & ( Xa2 = one_one(nat) ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw)),Xa2)) )
           => ( ! [V2: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy,Uz2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy,Uz2)),Xa2)) )
             => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xa2)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),Xa2))
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                               => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                                  & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                                       => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_1827_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_V5719532721284313246member(X,Xa2)
      <=> pp(Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A5: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A5,B4) )
             => ( ( pp(Y)
                <=> ( ( ( Xa2 = zero_zero(nat) )
                     => pp(A5) )
                    & ( ( Xa2 != zero_zero(nat) )
                     => ( ( ( Xa2 = one_one(nat) )
                         => pp(B4) )
                        & ( Xa2 = one_one(nat) ) ) ) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),Xa2)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw) )
               => ( ~ pp(Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw)),Xa2)) ) )
           => ~ ! [Uy: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy,aa(nat,nat,suc,V2),TreeList,S) )
                 => ( ( pp(Y)
                    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                         => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) )
                   => ~ 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,V2),TreeList,S)),Xa2)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_1828_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_V5719532721284313246member(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A5: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A5,B4) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),Xa2))
               => ~ ( ( ( Xa2 = zero_zero(nat) )
                     => pp(A5) )
                    & ( ( Xa2 != zero_zero(nat) )
                     => ( ( ( Xa2 = one_one(nat) )
                         => pp(B4) )
                        & ( Xa2 = one_one(nat) ) ) ) ) ) )
         => ~ ! [Uy: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy,aa(nat,nat,suc,V2),TreeList,S) )
               => ( 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,V2),TreeList,S)),Xa2))
                 => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_1829_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A5: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A5,B4) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),Xa2))
               => ( ( ( Xa2 = zero_zero(nat) )
                   => pp(A5) )
                  & ( ( Xa2 != zero_zero(nat) )
                   => ( ( ( Xa2 = one_one(nat) )
                       => pp(B4) )
                      & ( Xa2 = one_one(nat) ) ) ) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw)),Xa2)) )
           => ~ ! [Uy: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy,aa(nat,nat,suc,V2),TreeList,S) )
                 => ( 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,V2),TreeList,S)),Xa2))
                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_1830_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( pp(aa(nat,bool,vEBT_vebt_member(X),Xa2))
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [A5: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A5,B4) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(A5,B4)),Xa2))
               => ~ ( ( ( Xa2 = zero_zero(nat) )
                     => pp(A5) )
                    & ( ( Xa2 != zero_zero(nat) )
                     => ( ( ( Xa2 = one_one(nat) )
                         => pp(B4) )
                        & ( Xa2 = one_one(nat) ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),Xa2))
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa2),Mi2))
                           => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),Xa2))
                               => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                                   => pp(aa(nat,bool,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) )
                                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_1831_zmod__numeral__Bit0,axiom,
    ! [V3: num,W2: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),bit0(V3)),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),V3),aa(num,int,numeral_numeral(int),W2))) ).

% zmod_numeral_Bit0
tff(fact_1832_zmod__le__nonneg__dividend,axiom,
    ! [M2: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,M2,K2)),M2)) ) ).

% zmod_le_nonneg_dividend
tff(fact_1833_zmod__eq__0__iff,axiom,
    ! [M2: int,D3: int] :
      ( ( modulo_modulo(int,M2,D3) = zero_zero(int) )
    <=> ? [Q5: int] : M2 = aa(int,int,aa(int,fun(int,int),times_times(int),D3),Q5) ) ).

% zmod_eq_0_iff
tff(fact_1834_zmod__eq__0D,axiom,
    ! [M2: int,D3: int] :
      ( ( modulo_modulo(int,M2,D3) = zero_zero(int) )
     => ? [Q3: int] : M2 = aa(int,int,aa(int,fun(int,int),times_times(int),D3),Q3) ) ).

% zmod_eq_0D
tff(fact_1835_zmod__int,axiom,
    ! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,A3,B2)) = modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zmod_int
tff(fact_1836_mod__int__unique,axiom,
    ! [K2: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( modulo_modulo(int,K2,L) = R ) ) ).

% mod_int_unique
tff(fact_1837_neg__mod__conj,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,A3,B2)),zero_zero(int)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),modulo_modulo(int,A3,B2))) ) ) ).

% neg_mod_conj
tff(fact_1838_pos__mod__conj,axiom,
    ! [B2: int,A3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,A3,B2)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),modulo_modulo(int,A3,B2)),B2)) ) ) ).

% pos_mod_conj
tff(fact_1839_zmod__trivial__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( modulo_modulo(int,I2,K2) = I2 )
    <=> ( ( K2 = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K2)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),I2)) ) ) ) ).

% zmod_trivial_iff
tff(fact_1840_div__mod__decomp__int,axiom,
    ! [A6: int,N: int] : A6 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),divide_divide(int,A6,N)),N)),modulo_modulo(int,A6,N)) ).

% div_mod_decomp_int
tff(fact_1841_eucl__rel__int,axiom,
    ! [K2: int,L: int] : eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,K2,L)),modulo_modulo(int,K2,L))) ).

% eucl_rel_int
tff(fact_1842_mod__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K2))
       => ( modulo_modulo(int,K2,L) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_1843_int__mod__pos__eq,axiom,
    ! [A3: int,B2: int,Q2: int,R: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),B2))
         => ( modulo_modulo(int,A3,B2) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1844_int__mod__neg__eq,axiom,
    ! [A3: int,B2: int,Q2: int,R: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R))
         => ( modulo_modulo(int,A3,B2) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1845_split__zmod,axiom,
    ! [P2: fun(int,bool),N: int,K2: int] :
      ( pp(aa(int,bool,P2,modulo_modulo(int,N,K2)))
    <=> ( ( ( K2 = zero_zero(int) )
         => pp(aa(int,bool,P2,N)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
         => ! [I: int,J: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),K2))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),J) ) )
             => pp(aa(int,bool,P2,J)) ) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
         => ! [I: int,J: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),J))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),zero_zero(int)))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),J) ) )
             => pp(aa(int,bool,P2,J)) ) ) ) ) ).

% split_zmod
tff(fact_1846_zmod__zmult2__eq,axiom,
    ! [C3: int,A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C3))
     => ( modulo_modulo(int,A3,aa(int,int,aa(int,fun(int,int),times_times(int),B2),C3)) = 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,divide_divide(int,A3,B2),C3))),modulo_modulo(int,A3,B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_1847_split__pos__lemma,axiom,
    ! [K2: int,P2: fun(int,fun(int,bool)),N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P2,divide_divide(int,N,K2)),modulo_modulo(int,N,K2)))
      <=> ! [I: int,J: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J))
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),K2))
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),J) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P2,I),J)) ) ) ) ).

% split_pos_lemma
tff(fact_1848_split__neg__lemma,axiom,
    ! [K2: int,P2: fun(int,fun(int,bool)),N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P2,divide_divide(int,N,K2)),modulo_modulo(int,N,K2)))
      <=> ! [I: int,J: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),J))
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),zero_zero(int)))
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),J) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P2,I),J)) ) ) ) ).

% split_neg_lemma
tff(fact_1849_pos__zmod__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,B2,A3))) ) ) ).

% pos_zmod_mult_2
tff(fact_1850_neg__zmod__mult__2,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(int,int,aa(int,fun(int,int),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)),A3))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_1851_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_VEBT_membermima(X,Xa2)
      <=> pp(Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ~ pp(Y)
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2)) ) )
         => ( ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy) )
               => ( ~ pp(Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)),Xa2)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va2,Vb2) )
                 => ( ( pp(Y)
                    <=> ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va2,Vb2)),Xa2)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList,Vc2) )
                   => ( ( pp(Y)
                      <=> ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                             => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList,Vc2)),Xa2)) ) )
               => ~ ! [V2: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd) )
                     => ( ( pp(Y)
                        <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                             => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) )
                       => ~ 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,V2),TreeList,Vd)),Xa2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_1852_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2)) )
         => ( ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy) )
               => ~ 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)),Xa2)) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va2,Vb2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va2,Vb2)),Xa2))
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList,Vc2) )
                   => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList,Vc2)),Xa2))
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) )
               => ~ ! [V2: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd)),Xa2))
                       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_1853_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_VEBT_membermima(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Mi2: nat,Ma2: nat,Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va2,Vb2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va2,Vb2)),Xa2))
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                ( ( X = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList,Vc2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V2),TreeList,Vc2)),Xa2))
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) )
           => ~ ! [V2: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd)),Xa2))
                   => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
                         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa2,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_1854_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( arcosh(A,one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1855_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z2: A,K5: real,N: nat] :
          ( ( H != zero_zero(A) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z2)),K5))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),H))),K5))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),H)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),N)),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),K5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H)))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_1856_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,N,divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_1857_artanh__0,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ( artanh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% artanh_0
tff(fact_1858_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K6))
              & ! [N5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),K6)) )
        <=> ? [N7: nat] :
            ! [N5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7)))) ) ) ).

% lemma_NBseq_def
tff(fact_1859_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K6))
              & ! [N5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),K6)) )
        <=> ? [N7: nat] :
            ! [N5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7)))) ) ) ).

% lemma_NBseq_def2
tff(fact_1860_norm__mult__numeral1,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [W2: num,A3: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),W2)),real_V7770717601297561774m_norm(A,A3)) ) ).

% norm_mult_numeral1
tff(fact_1861_norm__mult__numeral2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A,W2: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A3)),aa(num,real,numeral_numeral(real),W2)) ) ).

% norm_mult_numeral2
tff(fact_1862_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,X)),zero_zero(real)))
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_1863_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,X)))
        <=> ( X != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_1864_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_1865_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_1866_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z2: A,W2: A,M2: nat] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z2)),one_one(real)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,W2)),one_one(real)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),W2),M2)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M2)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z2),W2))))) ) ) ) ).

% norm_power_diff
tff(fact_1867_norm__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,Y: A] : 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
tff(fact_1868_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,divide_divide(A,A3,B2)) = divide_divide(real,real_V7770717601297561774m_norm(A,A3),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_1869_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W2: A,N: nat,Z2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W2),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),N) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( real_V7770717601297561774m_norm(A,W2) = real_V7770717601297561774m_norm(A,Z2) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_1870_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,R: real,Y: A,S2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),R))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S2))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R),S2))) ) ) ) ).

% norm_mult_less
tff(fact_1871_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,Y: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),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_1872_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W2: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W2),N) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W2) = one_one(real) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_1873_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( arsinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_1874_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
                     => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) )
                    & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
                     => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_1875_prod_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),X: fun(B,A),Y: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ap(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ap(set(B),fun(fun(B,A),fun(B,bool)),I5),Y))))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_aq(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y)))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_1876_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [N: nat,M2: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( ( ( X = one_one(A) )
               => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),M2)) ) )
              & ( ( X != one_one(A) )
               => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ) ) ).

% sum_gp
tff(fact_1877_sum_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),X: fun(B,A),Y: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I5),Y))))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_as(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y)))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_1878_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),one_one(real)))
         => sums(A,aTP_Lamp_at(A,fun(nat,A),Z2),divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_1879_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z2: A,N: 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))),N)) = 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,aa(nat,fun(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))),N)))),comm_s3205402744901411588hammer(A,Z2,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),N)) ) ).

% pochhammer_double
tff(fact_1880_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),zero_zero(A))) ) ).

% dvd_0_right
tff(fact_1881_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),A3))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_1882_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_triv_right_iff
tff(fact_1883_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_triv_left_iff
tff(fact_1884_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),divide_divide(A,B2,A3)),divide_divide(A,C3,A3)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C3)) ) ) ) ) ).

% div_dvd_div
tff(fact_1885_nat__dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),one_one(nat)))
    <=> ( M2 = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_1886_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_au(B,A)),A6) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_1887_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C3)) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1888_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C3)) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1889_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
        <=> ( ( C3 = zero_zero(A) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_1890_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
        <=> ( ( C3 = zero_zero(A) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_1891_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1892_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1893_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A))) ) ) ) ).

% unit_prod
tff(fact_1894_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,A3)),A3) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_1895_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,A3)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_1896_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,C3)),divide_divide(A,B2,C3)) ) ) ) ) ).

% div_add
tff(fact_1897_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),divide_divide(A,A3,B2)),one_one(A))) ) ) ) ).

% unit_div
tff(fact_1898_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),divide_divide(A,one_one(A),A3)),one_one(A))) ) ) ).

% unit_div_1_unit
tff(fact_1899_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( divide_divide(A,one_one(A),divide_divide(A,one_one(A),A3)) = A3 ) ) ) ).

% unit_div_1_div_1
tff(fact_1900_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_1901_sum_Oinfinite,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),G3: fun(B,A)] :
          ( ~ pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A6) = zero_zero(A) ) ) ) ).

% sum.infinite
tff(fact_1902_sum__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [F4: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),F4))
         => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),F4) = zero_zero(A) )
          <=> ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),F4))
               => ( aa(B,A,F3,X4) = zero_zero(A) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_1903_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,C3)),divide_divide(A,B2,C3)) ) ) ) ) ).

% div_diff
tff(fact_1904_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( modulo_modulo(A,B2,A3) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_1905_dvd__1__left,axiom,
    ! [K2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,suc,zero_zero(nat))),K2)) ).

% dvd_1_left
tff(fact_1906_dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( M2 = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_1907_nat__mult__dvd__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
    <=> ( ( K2 = zero_zero(nat) )
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1908_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,zero_zero(nat)) = one_one(A) ) ).

% pochhammer_0
tff(fact_1909_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% pochhammer_Suc0
tff(fact_1910_sum_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( 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_av(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( 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_av(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = zero_zero(A) ) ) ) ) ) ).

% sum.delta
tff(fact_1911_sum_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( 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_aw(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( 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_aw(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = zero_zero(A) ) ) ) ) ) ).

% sum.delta'
tff(fact_1912_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,one_one(A),A3)) = divide_divide(A,B2,A3) ) ) ) ).

% unit_mult_div_div
tff(fact_1913_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,A3)),A3) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_1914_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [N: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% pow_divides_pow_iff
tff(fact_1915_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A6: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ax(A,fun(B,A),Y)),A6) = 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),A6))),Y) ) ).

% sum_constant
tff(fact_1916_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)) ) ) ) ).

% even_mult_iff
tff(fact_1917_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,N))))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% even_Suc_Suc_iff
tff(fact_1918_even__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% even_Suc
tff(fact_1919_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: fun(nat,A),X: A] :
          ( sums(A,aTP_Lamp_ay(fun(nat,A),fun(nat,A),A3),X)
        <=> ( aa(nat,A,A3,zero_zero(nat)) = X ) ) ) ).

% powser_sums_zero_iff
tff(fact_1920_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( divide_divide(nat,aa(nat,nat,suc,N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ).

% even_Suc_div_two
tff(fact_1921_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( divide_divide(nat,aa(nat,nat,suc,N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_1922_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M2: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_1923_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A6: set(nat),C3: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A6)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_az(fun(nat,A),fun(nat,A),C3)),A6) = aa(nat,A,C3,zero_zero(nat)) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
                & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A6)) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_az(fun(nat,A),fun(nat,A),C3)),A6) = zero_zero(A) ) ) ) ) ).

% sum_zero_power
tff(fact_1924_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W2))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1925_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% even_power
tff(fact_1926_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% power_less_zero_eq
tff(fact_1927_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1928_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_1929_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A6: set(nat),C3: fun(nat,A),D3: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A6)) )
           => ( 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_ba(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C3),D3)),A6) = divide_divide(A,aa(nat,A,C3,zero_zero(nat)),aa(nat,A,D3,zero_zero(nat))) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
                & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A6)) )
           => ( 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_ba(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C3),D3)),A6) = zero_zero(A) ) ) ) ) ).

% sum_zero_power'
tff(fact_1930_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A3 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_1931_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A))))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_1932_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W2))))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W2) = zero_zero(nat) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
              & ( A3 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1933_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_1934_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W2)))
            & ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
              | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1935_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = divide_divide(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_1936_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_1937_sum_Oneutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),G3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
             => ( aa(B,A,G3,X3) = zero_zero(A) ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A6) = zero_zero(A) ) ) ) ).

% sum.neutral
tff(fact_1938_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),A6: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A6) != zero_zero(A) )
         => ~ ! [A5: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),A6))
               => ( aa(B,A,G3,A5) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_1939_sum__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [K5: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),K5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(B,A,G3,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),K5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),K5))) ) ) ).

% sum_mono
tff(fact_1940_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R: A,F3: fun(B,A),A6: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A6)) = 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_bb(A,fun(fun(B,A),fun(B,A)),R),F3)),A6) ) ).

% sum_distrib_left
tff(fact_1941_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F3: fun(B,A),A6: set(B),R: 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),F3),A6)),R) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_bc(fun(B,A),fun(A,fun(B,A)),F3),R)),A6) ) ).

% sum_distrib_right
tff(fact_1942_sum__product,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_0(B)
     => ! [F3: fun(A,B),A6: set(A),G3: fun(C,B),B6: set(C)] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A6)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G3),B6)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_be(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),F3),G3),B6)),A6) ) ).

% sum_product
tff(fact_1943_dvd__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M2))
       => ( M2 = N ) ) ) ).

% dvd_antisym
tff(fact_1944_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3)) ) ) ) ).

% dvd_trans
tff(fact_1945_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),A3)) ) ).

% dvd_refl
tff(fact_1946_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
        <=> ( ( A3 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_1947_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),A3))
         => ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_1948_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),P),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
         => ~ ! [X3: A,Y3: A] :
                ( ( P = aa(A,A,aa(A,fun(A,A),times_times(A),X3),Y3) )
               => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X3),A3))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y3),B2)) ) ) ) ) ).

% dvd_productE
tff(fact_1949_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
         => ? [B10: A,C6: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B10),C6) )
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B10),B2))
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C6),C3)) ) ) ) ).

% division_decomp
tff(fact_1950_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ~ ! [K: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) ) ) ).

% dvdE
tff(fact_1951_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A3: A,B2: A,K2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% dvdI
tff(fact_1952_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
        <=> ? [K3: A] : A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K3) ) ) ).

% dvd_def
tff(fact_1953_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ).

% dvd_mult
tff(fact_1954_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))) ) ) ).

% dvd_mult2
tff(fact_1955_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3)) ) ) ).

% dvd_mult_left
tff(fact_1956_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ).

% dvd_triv_left
tff(fact_1957_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3))) ) ) ) ).

% mult_dvd_mono
tff(fact_1958_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C3)) ) ) ).

% dvd_mult_right
tff(fact_1959_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3))) ) ).

% dvd_triv_right
tff(fact_1960_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3)) ) ) ) ).

% dvd_add_right_iff
tff(fact_1961_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% dvd_add_left_iff
tff(fact_1962_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))) ) ) ) ).

% dvd_add
tff(fact_1963_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_1964_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% unit_imp_dvd
tff(fact_1965_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),one_one(A)),A3)) ) ).

% one_dvd
tff(fact_1966_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2))) ) ) ) ).

% dvd_diff
tff(fact_1967_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( divide_divide(A,divide_divide(A,A3,D3),divide_divide(A,B2,D3)) = divide_divide(A,A3,B2) ) ) ) ) ).

% div_div_div_same
tff(fact_1968_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,C3: A,B2: A] :
          ( ( divide_divide(A,A3,C3) = divide_divide(A,B2,C3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
             => ( A3 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1969_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
           => ( ( divide_divide(A,A3,C3) = divide_divide(A,B2,C3) )
            <=> ( A3 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1970_gcd__nat_Oextremum,axiom,
    ! [A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),zero_zero(nat))) ).

% gcd_nat.extremum
tff(fact_1971_gcd__nat_Oextremum__strict,axiom,
    ! [A3: nat] :
      ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A3))
        & ( zero_zero(nat) != A3 ) ) ).

% gcd_nat.extremum_strict
tff(fact_1972_gcd__nat_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A3))
    <=> ( A3 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_1973_gcd__nat_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),zero_zero(nat)))
        & ( A3 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_1974_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A3))
     => ( A3 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_1975_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),modulo_modulo(A,A3,B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3)) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1976_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),modulo_modulo(A,A3,B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3)) ) ) ) ).

% dvd_mod_iff
tff(fact_1977_sum__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A6))) ) ) ).

% sum_nonneg
tff(fact_1978_sum__nonpos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A6)),zero_zero(A))) ) ) ).

% sum_nonpos
tff(fact_1979_dvd__diff__nat,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N))) ) ) ).

% dvd_diff_nat
tff(fact_1980_sum__mono__inv,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F3: fun(I6,A),I5: set(I6),G3: fun(I6,A),I2: I6] :
          ( ( aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),F3),I5) = aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),G3),I5) )
         => ( ! [I3: I6] :
                ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I3),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I6,A,F3,I3)),aa(I6,A,G3,I3))) )
           => ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I2),I5))
             => ( pp(aa(set(I6),bool,finite_finite2(I6),I5))
               => ( aa(I6,A,F3,I2) = aa(I6,A,G3,I2) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_1981_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(nat),F3: fun(nat,A),G3: fun(nat,A)] :
          ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A6))
         => ( ! [X3: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,X3)),A6))
               => ( aa(nat,A,F3,aa(nat,nat,suc,X3)) = aa(nat,A,G3,aa(nat,nat,suc,X3)) ) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),A6) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),A6) ) ) ) ) ).

% sum_cong_Suc
tff(fact_1982_sum_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),G3: fun(B,A),P2: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(fun(B,bool),set(B),collect(B),aa(fun(B,bool),fun(B,bool),aTP_Lamp_bf(set(B),fun(fun(B,bool),fun(B,bool)),A6),P2))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_bg(fun(B,A),fun(fun(B,bool),fun(B,A)),G3),P2)),A6) ) ) ) ).

% sum.inter_filter
tff(fact_1983_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bh(A,fun(A,bool),A3))),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bh(A,fun(A,bool),B2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% subset_divisors_dvd
tff(fact_1984_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bh(A,fun(A,bool),A3))),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bh(A,fun(A,bool),B2))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_1985_sum__le__included,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S2: set(B),T2: set(C),G3: fun(C,A),I2: fun(C,B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( pp(aa(set(C),bool,finite_finite2(C),T2))
           => ( ! [X3: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),T2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(C,A,G3,X3))) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
                   => ? [Xa: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Xa),T2))
                        & ( aa(C,B,I2,Xa) = X3 )
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(C,A,G3,Xa))) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),S2)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G3),T2))) ) ) ) ) ) ).

% sum_le_included
tff(fact_1986_sum__nonneg__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A6) = zero_zero(A) )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A6))
                 => ( aa(B,A,F3,X4) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_1987_sum__strict__mono__ex1,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A6: set(I6),F3: fun(I6,A),G3: fun(I6,A)] :
          ( pp(aa(set(I6),bool,finite_finite2(I6),A6))
         => ( ! [X3: I6] :
                ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I6,A,F3,X3)),aa(I6,A,G3,X3))) )
           => ( ? [X5: I6] :
                  ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),X5),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(I6,A,F3,X5)),aa(I6,A,G3,X5))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),F3),A6)),aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7311177749621191930dd_sum(I6,A),G3),A6))) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_1988_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R2: fun(A,fun(A,bool)),S3: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R2,zero_zero(A)),zero_zero(A)))
         => ( ! [X15: A,Y15: A,X23: A,Y23: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X15),X23))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R2,Y15),Y23)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X15),Y15)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X23),Y23))) )
           => ( pp(aa(set(B),bool,finite_finite2(B),S3))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(B,A,H,X3)),aa(B,A,G3,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),S3))) ) ) ) ) ) ).

% sum.related
tff(fact_1989_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S5: set(B),T4: set(C),S3: set(B),I2: fun(C,B),J2: fun(B,C),T5: set(C),G3: fun(B,A),H: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T4))
           => ( ! [A5: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S5)))
                 => ( aa(C,B,I2,aa(B,C,J2,A5)) = A5 ) )
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S5)))
                   => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,J2,A5)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4))) )
               => ( ! [B4: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4)))
                     => ( aa(B,C,J2,aa(C,B,I2,B4)) = B4 ) )
                 => ( ! [B4: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4)))
                       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(C,B,I2,B4)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S5))) )
                   => ( ! [A5: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),S5))
                         => ( aa(B,A,G3,A5) = zero_zero(A) ) )
                     => ( ! [B4: C] :
                            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),T4))
                           => ( aa(C,A,H,B4) = zero_zero(A) ) )
                       => ( ! [A5: B] :
                              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),S3))
                             => ( aa(C,A,H,aa(B,C,J2,A5)) = aa(B,A,G3,A5) ) )
                         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),S3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),H),T5) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_1990_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),one_one(A))) ) ).

% not_is_unit_0
tff(fact_1991_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( ( divide_divide(A,A3,B2) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1992_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) )
          <=> ( B2 = C3 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_1993_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) )
          <=> ( B2 = C3 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_1994_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C3)) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1995_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3)) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1996_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3)) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_1997_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3)) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_1998_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A))) ) ) ) ).

% is_unit_mult_iff
tff(fact_1999_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C3)),A3) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3),C3) ) ) ) ).

% dvd_div_mult
tff(fact_2000_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) ) ) ) ).

% div_mult_swap
tff(fact_2001_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( divide_divide(A,A3,divide_divide(A,B2,C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),C3) ) ) ) ) ).

% div_div_eq_right
tff(fact_2002_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),A3))
         => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A3,B2),C3) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_2003_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),divide_divide(A,B2,C3))) ) ) ).

% dvd_mult_imp_div
tff(fact_2004_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,D3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),C3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),divide_divide(A,C3,D3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2005_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( ( divide_divide(A,B2,A3) = divide_divide(A,C3,A3) )
          <=> ( B2 = C3 ) ) ) ) ).

% unit_div_cancel
tff(fact_2006_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),divide_divide(A,A3,B2)),C3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3)) ) ) ) ).

% div_unit_dvd_iff
tff(fact_2007_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),divide_divide(A,C3,B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3)) ) ) ) ).

% dvd_div_unit_iff
tff(fact_2008_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A] :
          ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_2009_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
        <=> ( modulo_modulo(A,B2,A3) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_2010_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] :
          ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% mod_0_imp_dvd
tff(fact_2011_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A,N: nat,M2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),M2))) ) ) ) ).

% dvd_power_le
tff(fact_2012_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat,B2: A,M2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),B2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),B2)) ) ) ) ).

% power_le_dvd
tff(fact_2013_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M2: nat,N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% le_imp_power_dvd
tff(fact_2014_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,A3,B2)))) ) ).

% dvd_minus_mod
tff(fact_2015_dvd__pos__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2)) ) ) ).

% dvd_pos_nat
tff(fact_2016_nat__dvd__not__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
       => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M2)) ) ) ).

% nat_dvd_not_less
tff(fact_2017_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_pos
tff(fact_2018_dvd__minus__self,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N)) ) ) ).

% dvd_minus_self
tff(fact_2019_sum__nonneg__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S2: set(B),F3: fun(B,A),I2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),S2) = zero_zero(A) )
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),S2))
               => ( aa(B,A,F3,I2) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_2020_sum__nonneg__leq__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S2: set(B),F3: fun(B,A),B6: A,I2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),S2) = B6 )
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),B6)) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_2021_less__eq__dvd__minus,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ).

% less_eq_dvd_minus
tff(fact_2022_dvd__diffD1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),M2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),N)) ) ) ) ).

% dvd_diffD1
tff(fact_2023_dvd__diffD,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),M2)) ) ) ) ).

% dvd_diffD
tff(fact_2024_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat,M2: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => ( comm_s3205402744901411588hammer(A,A3,M2) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_2025_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M2: nat,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,M2) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => ( comm_s3205402744901411588hammer(A,A3,N) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_2026_bezout__add__nat,axiom,
    ! [A3: nat,B2: nat] :
    ? [D2: nat,X3: nat,Y3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A3))
      & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),B2))
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),D2) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_2027_bezout__lemma__nat,axiom,
    ! [D3: nat,A3: nat,B2: nat,X: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),B2))
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),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),A3),Y)),D3) ) )
         => ? [X3: nat,Y3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),A3))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),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),A3),B2)),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)),D3) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_2028_bezout1__nat,axiom,
    ! [A3: nat,B2: nat] :
    ? [D2: nat,X3: nat,Y3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A3))
      & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),B2))
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = D2 )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_2029_sum_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),G3: fun(B,A),B6: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(B),fun(B,A),aTP_Lamp_bi(fun(B,A),fun(set(B),fun(B,A)),G3),B6)),A6) ) ) ) ).

% sum.inter_restrict
tff(fact_2030_sum_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_bj(fun(B,A),fun(B,bool),G3)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A6) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_2031_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bk(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_2032_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M2: nat,K2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_bl(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_2033_sum__pos2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),I2: B,F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F3,I2)))
             => ( ! [I3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),I5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),I5))) ) ) ) ) ) ).

% sum_pos2
tff(fact_2034_sum__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( ( I5 != bot_bot(set(B)) )
           => ( ! [I3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),I5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F3,I3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),I5))) ) ) ) ) ).

% sum_pos
tff(fact_2035_sum__bounded__below,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A6: set(B),K5: A,F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K5),aa(B,A,F3,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A6))),K5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A6))) ) ) ).

% sum_bounded_below
tff(fact_2036_sum__bounded__above,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A6: set(B),F3: fun(B,A),K5: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),K5)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A6)),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),A6))),K5))) ) ) ).

% sum_bounded_above
tff(fact_2037_sum_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T5: set(B),S3: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,G3,X3) = zero_zero(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S3))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_2038_sum_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T5: set(B),S3: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,H,X3) = zero_zero(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S3))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T5) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_2039_sum_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T5: set(B),S3: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,G3,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),S3) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_2040_sum_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T5: set(B),S3: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,G3,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),T5) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_2041_sum_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C5: set(B),A6: set(B),B6: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),C5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),C5))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),A6)))
                   => ( aa(B,A,G3,A5) = zero_zero(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),B6)))
                     => ( aa(B,A,H,B4) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),C5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),C5) )
                   => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),B6) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_2042_sum_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C5: set(B),A6: set(B),B6: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),C5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),C5))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),A6)))
                   => ( aa(B,A,G3,A5) = zero_zero(A) ) )
               => ( ! [B4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),B6)))
                     => ( aa(B,A,H,B4) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),B6) )
                  <=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),C5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),C5) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_2043_sum_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T5: set(B),S3: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T5))
         => ( pp(aa(set(B),bool,finite_finite2(B),S3))
           => ( ! [I3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
                 => ( aa(B,A,H,I3) = zero_zero(A) ) )
             => ( ! [I3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),T5)))
                   => ( aa(B,A,G3,I3) = zero_zero(A) ) )
               => ( ! [X3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S3),T5)))
                     => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T5) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_2044_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P2: fun(A,bool),L: A] :
          ( ? [X4: A] : pp(aa(A,bool,P2,aa(A,A,aa(A,fun(A,A),times_times(A),L),X4)))
        <=> ? [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),zero_zero(A))))
              & pp(aa(A,bool,P2,X4)) ) ) ) ).

% unity_coeff_ex
tff(fact_2045_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [C2: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) ) ) ).

% unit_dvdE
tff(fact_2046_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C3: A,B2: A,D3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( C3 != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),D3))
               => ( ( divide_divide(A,B2,A3) = divide_divide(A,D3,C3) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2047_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A3: A] :
          ( ( C3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),divide_divide(A,B2,C3)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2)) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2048_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C3: A] :
          ( ( B2 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),divide_divide(A,A3,B2)),C3))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2049_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
           => ( ( divide_divide(A,B2,A3) = C3 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2050_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( ( divide_divide(A,A3,B2) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2051_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D5: A,T2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),D5))
         => ! [X5: A,K4: A] :
              ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),T2)))
            <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),T2))) ) ) ) ).

% inf_period(4)
tff(fact_2052_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D5: A,T2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),D5))
         => ! [X5: A,K4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),T2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),T2))) ) ) ) ).

% inf_period(3)
tff(fact_2053_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( ( divide_divide(A,A3,B2) = C3 )
          <=> ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_2054_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( ( A3 = divide_divide(A,C3,B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = C3 ) ) ) ) ).

% unit_eq_div2
tff(fact_2055_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A3,B2),C3) ) ) ) ) ).

% div_mult_unit2
tff(fact_2056_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),C3) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),B2) ) ) ) ).

% unit_div_commute
tff(fact_2057_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) ) ) ) ).

% unit_div_mult_swap
tff(fact_2058_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A)))
           => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A3,B2),C3) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2059_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
            | ( N = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_2060_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( modulo_modulo(A,A3,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_2061_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_nonneg
tff(fact_2062_dvd__imp__le,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N)) ) ) ).

% dvd_imp_le
tff(fact_2063_dvd__mult__cancel,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N)) ) ) ).

% dvd_mult_cancel
tff(fact_2064_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2065_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = zero_zero(A) ) ) ) ) ).

% pochhammer_0_left
tff(fact_2066_bezout__add__strong__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [D2: nat,X3: nat,Y3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A3))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),B2))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_2067_mod__greater__zero__iff__not__dvd,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M2,N)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M2)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_2068_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M2: nat,Q2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( ( modulo_modulo(nat,M2,Q2) = modulo_modulo(nat,N,Q2) )
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2069_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,M2: 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_bm(A,fun(nat,fun(nat,A)),X),M2)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),I5)) ) ).

% sum_power_add
tff(fact_2070_sum__mono2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [B6: set(B),A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),B6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B6))
           => ( ! [B4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B6),A6)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,B4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),B6))) ) ) ) ) ).

% sum_mono2
tff(fact_2071_sum_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),B6: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B6)))
                 => ( aa(B,A,G3,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),B6)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_2072_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,N,M2)) = 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_bn(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M2)),set_or1337092689740270186AtMost(nat,N,M2)) ) ).

% sum.atLeastAtMost_rev
tff(fact_2073_finite__divisors__nat,axiom,
    ! [M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_bo(nat,fun(nat,bool),M2)))) ) ).

% finite_divisors_nat
tff(fact_2074_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: fun(nat,A)] : sums(A,aTP_Lamp_ay(fun(nat,A),fun(nat,A),A3),aa(nat,A,A3,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_2075_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,zero_zero(nat),K2)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_2076_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_2077_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_2078_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_2079_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),zero_zero(A))) ) ).

% even_zero
tff(fact_2080_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [B4: A] :
                  ( ( B4 != zero_zero(A) )
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B4),one_one(A)))
                   => ( ( divide_divide(A,one_one(A),A3) = B4 )
                     => ( ( divide_divide(A,one_one(A),B4) = A3 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B4) = one_one(A) )
                         => ( divide_divide(A,C3,A3) != aa(A,A,aa(A,fun(A,A),times_times(A),C3),B4) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_2081_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = divide_divide(A,one_one(A),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2082_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) = divide_divide(A,one_one(A),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2083_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ~ ! [B4: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B4) ) ) ).

% evenE
tff(fact_2084_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,M2: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)))
          <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),one_one(A)))
              | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ) ).

% dvd_power_iff
tff(fact_2085_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            | ( X = one_one(A) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))) ) ) ).

% dvd_power
tff(fact_2086_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B6: set(A),A6: set(A),B2: A,F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B6))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B6),A6)))
             => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),zero_zero(B)),aa(A,B,F3,B2)))
               => ( ! [X3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B6))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X3))) )
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A6)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),B6))) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_2087_member__le__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I2: C,A6: set(C),F3: fun(C,B)] :
          ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),I2),A6))
         => ( ! [X3: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),A6),aa(set(C),set(C),insert(C,I2),bot_bot(set(C))))))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(C,B,F3,X3))) )
           => ( pp(aa(set(C),bool,finite_finite2(C),A6))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(C,B,F3,I2)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F3),A6))) ) ) ) ) ).

% member_le_sum
tff(fact_2088_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( divide_divide(nat,X,aa(num,nat,numeral_numeral(nat),bit0(one2))) = divide_divide(nat,Y,aa(num,nat,numeral_numeral(nat),bit0(one2))) )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Y)) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_2089_sum__bounded__above__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & semiring_1(A) )
     => ! [A6: set(B),F3: fun(B,A),K5: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,I3)),K5)) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),A6)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A6)),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),A6))),K5))) ) ) ) ).

% sum_bounded_above_strict
tff(fact_2090_sum__bounded__above__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_field(A)
     => ! [A6: set(B),F3: fun(B,A),K5: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),divide_divide(A,K5,aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A6))))) )
         => ( pp(aa(set(B),bool,finite_finite2(B),A6))
           => ( ( A6 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A6)),K5)) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_2091_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),N)) ) ).

% pochhammer_rec
tff(fact_2092_dvd__mult__cancel1,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),M2))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_2093_dvd__mult__cancel2,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M2)),M2))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_2094_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z2: A,N: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,N)) = 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),N))),comm_s3205402744901411588hammer(A,Z2,N)) ) ).

% pochhammer_rec'
tff(fact_2095_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A3,N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% pochhammer_Suc
tff(fact_2096_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),M2)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Q2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),M2)),Q2)))) ) ) ) ).

% dvd_minus_add
tff(fact_2097_power__dvd__imp__le,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% power_dvd_imp_le
tff(fact_2098_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z2: A,N: nat,M2: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,N)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),N)),M2)) ) ).

% pochhammer_product'
tff(fact_2099_mod__nat__eqI,axiom,
    ! [R: nat,N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),R),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R),M2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),R)))
         => ( modulo_modulo(nat,M2,N) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_2100_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N))),aa(nat,A,G3,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bk(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_2101_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bp(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,aa(nat,nat,suc,N))),aa(nat,A,F3,M2)) ) ) ) ).

% sum_Suc_diff
tff(fact_2102_sum__norm__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [S3: set(B),F3: fun(B,A),K5: real] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S3))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(B,A,F3,X3))),K5)) )
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),S3))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(set(B),nat,finite_card(B),S3))),K5))) ) ) ).

% sum_norm_bound
tff(fact_2103_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A),P: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_2104_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) = A3 ) ) ) ).

% even_two_times_div_two
tff(fact_2105_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_2106_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A,B2: A] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono_odd
tff(fact_2107_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% odd_pos
tff(fact_2108_dvd__power__iff__le,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K2),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% dvd_power_iff_le
tff(fact_2109_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,M2,A3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
            | ( M2 = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_2110_set__encode__def,axiom,
    nat_set_encode = aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% set_encode_def
tff(fact_2111_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,M2,A3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
            & ( M2 != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_2112_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,M2,A3)))
        <=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
            <=> ( M2 = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_2113_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M2: nat,N: nat,Z2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( comm_s3205402744901411588hammer(A,Z2,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,M2)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),M2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).

% pochhammer_product
tff(fact_2114_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ~ ! [B4: A] : A3 != 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_2115_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
           => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
           => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ) ).

% mod2_eq_if
tff(fact_2116_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
           => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
         => ~ ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
             => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_2117_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% zero_le_even_power
tff(fact_2118_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% zero_le_odd_power
tff(fact_2119_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_le_power_eq
tff(fact_2120_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,F3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bq(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,M2)),aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bq(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,M2,N)) = zero_zero(A) ) ) ) ) ).

% sum_natinterval_diff
tff(fact_2121_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_br(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,N)),aa(nat,A,F3,M2)) ) ) ) ).

% sum_telescope''
tff(fact_2122_even__set__encode__iff,axiom,
    ! [A6: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(nat),nat,nat_set_encode,A6)))
      <=> ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A6)) ) ) ).

% even_set_encode_iff
tff(fact_2123_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
        <=> ( ( N = zero_zero(nat) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
              & ( A3 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_less_power_eq
tff(fact_2124_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M2: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ) ) ).

% sum_gp_multiplied
tff(fact_2125_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2),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(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bs(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% sum.in_pairs
tff(fact_2126_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ).

% even_mask_div_iff'
tff(fact_2127_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            & ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
              | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2128_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_bt(nat,fun(nat,bool)),N))) ).

% mask_eq_sum_exp_nat
tff(fact_2129_gauss__sum__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_bu(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% gauss_sum_nat
tff(fact_2130_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% even_mod_4_div_2
tff(fact_2131_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ).

% even_mask_div_iff
tff(fact_2132_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_2133_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,D3: A,N: 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_bv(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D3))) ) ).

% double_arith_series
tff(fact_2134_arith__series__nat,axiom,
    ! [A3: nat,D3: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_bw(nat,fun(nat,fun(nat,nat)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),D3))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_2135_Sum__Icc__nat,axiom,
    ! [M2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_bu(nat,nat)),set_or1337092689740270186AtMost(nat,M2,N)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Icc_nat
tff(fact_2136_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => pp(aa(real,bool,aa(real,fun(real,bool),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),N)),X))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),N))) ) ).

% Bernoulli_inequality_even
tff(fact_2137_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum
tff(fact_2138_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,D3: A,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_bx(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D3))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_2139_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_2140_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M2: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_2141_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M2: nat,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp_offset
tff(fact_2142_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va3: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va3))))
       => ( vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va3))) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va3))))
       => ( vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va3))) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_2143_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_2144_power__half__series,axiom,
    sums(real,aTP_Lamp_by(nat,real),one_one(real)) ).

% power_half_series
tff(fact_2145_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => sums(A,aTP_Lamp_bz(nat,A),zero_zero(A)) ) ).

% sums_zero
tff(fact_2146_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [M2: nat,Z2: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ca(nat,fun(A,fun(nat,A)),M2),Z2),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),M2)) ) ).

% powser_sums_if
tff(fact_2147_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A6: set(nat),F3: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cb(set(nat),fun(fun(nat,A),fun(nat,A)),A6),F3),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),A6)) ) ) ).

% sums_If_finite_set
tff(fact_2148_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P2: fun(nat,bool),F3: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),P2)))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cc(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),P2),F3),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(fun(nat,bool),set(nat),collect(nat),P2))) ) ) ).

% sums_If_finite
tff(fact_2149_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N6: set(nat),F3: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),N6))
         => ( ! [N3: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N3),N6))
               => ( aa(nat,A,F3,N3) = zero_zero(A) ) )
           => sums(A,F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),N6)) ) ) ) ).

% sums_finite
tff(fact_2150_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [N: nat,F3: fun(nat,A),S2: A] :
          ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
             => ( aa(nat,A,F3,I3) = zero_zero(A) ) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cd(nat,fun(fun(nat,A),fun(nat,A)),N),F3),S2)
          <=> sums(A,F3,S2) ) ) ) ).

% sums_zero_iff_shift
tff(fact_2151_zdvd__mono,axiom,
    ! [K2: int,M2: int,T2: int] :
      ( ( K2 != zero_zero(int) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M2),T2))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),T2))) ) ) ).

% zdvd_mono
tff(fact_2152_zdvd__mult__cancel,axiom,
    ! [K2: int,M2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),N)))
     => ( ( K2 != zero_zero(int) )
       => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M2),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_2153_zdvd__period,axiom,
    ! [A3: int,D3: int,X: int,T2: int,C3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),D3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),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),C3),D3))),T2))) ) ) ).

% zdvd_period
tff(fact_2154_zdvd__reduce,axiom,
    ! [K2: int,N: int,M2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K2),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),aa(int,int,aa(int,fun(int,int),times_times(int),K2),M2))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K2),N)) ) ).

% zdvd_reduce
tff(fact_2155_sum__subtractf__nat,axiom,
    ! [A: $tType,A6: set(A),G3: fun(A,nat),F3: fun(A,nat)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,G3,X3)),aa(A,nat,F3,X3))) )
     => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_ce(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G3),F3)),A6) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A6)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G3),A6)) ) ) ).

% sum_subtractf_nat
tff(fact_2156_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A6) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
            & ( aa(A,nat,F3,X4) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A6))
               => ( ( X4 != Xa3 )
                 => ( aa(A,nat,F3,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_2157_sum__SucD,axiom,
    ! [A: $tType,F3: fun(A,nat),A6: set(A),N: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A6) = aa(nat,nat,suc,N) )
     => ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X3))) ) ) ).

% sum_SucD
tff(fact_2158_sum__eq__1__iff,axiom,
    ! [A: $tType,A6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A6) = one_one(nat) )
      <=> ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
            & ( aa(A,nat,F3,X4) = one_one(nat) )
            & ! [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A6))
               => ( ( X4 != Xa3 )
                 => ( aa(A,nat,F3,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_2159_sum__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),A6: set(A)] : aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_cf(fun(A,nat),fun(A,nat),F3)),A6) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A6)),aa(set(A),nat,finite_card(A),A6)) ).

% sum_Suc
tff(fact_2160_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S3: set(A),T5: set(B),R2: fun(A,fun(B,bool)),K2: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),S3))
     => ( pp(aa(set(B),bool,finite_finite2(B),T5))
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),T5))
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_cg(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),S3),R2),X3))) = K2 ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_ci(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),T5),R2)),S3) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(set(B),nat,finite_card(B),T5)) ) ) ) ) ).

% sum_multicount
tff(fact_2161_mod__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K2,L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K2))
        | ( ( L = zero_zero(int) )
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2)) )
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L)) ) ) ).

% mod_int_pos_iff
tff(fact_2162_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),X6))
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xs)),X6) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_2163_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,A),S2: A,T2: A] :
          ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N3)),aa(nat,A,G3,N3)))
         => ( sums(A,F3,S2)
           => ( sums(A,G3,T2)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),S2),T2)) ) ) ) ) ).

% sums_le
tff(fact_2164_sums__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A)] :
          ( ! [N3: nat] : aa(nat,A,F3,N3) = zero_zero(A)
         => sums(A,F3,zero_zero(A)) ) ) ).

% sums_0
tff(fact_2165_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I2: nat,F3: fun(nat,A)] : sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cj(nat,fun(fun(nat,A),fun(nat,A)),I2),F3),aa(nat,A,F3,I2)) ) ).

% sums_single
tff(fact_2166_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(nat,A),A3: A,C3: A] :
          ( sums(A,F3,A3)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),F3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ).

% sums_mult2
tff(fact_2167_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(nat,A),A3: A,C3: A] :
          ( sums(A,F3,A3)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,A)),F3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)) ) ) ).

% sums_mult
tff(fact_2168_Sum__Icc__int,axiom,
    ! [M2: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M2),N))
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_cm(int,int)),set_or1337092689740270186AtMost(int,M2,N)) = divide_divide(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M2),aa(int,int,aa(int,fun(int,int),minus_minus(int),M2),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).

% Sum_Icc_int
tff(fact_2169_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C3: A,F3: fun(nat,A),D3: A] :
          ( ( C3 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cn(A,fun(fun(nat,A),fun(nat,A)),C3),F3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D3))
          <=> sums(A,F3,D3) ) ) ) ).

% sums_mult_iff
tff(fact_2170_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C3: A,F3: fun(nat,A),D3: A] :
          ( ( C3 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_co(A,fun(fun(nat,A),fun(nat,A)),C3),F3),aa(A,A,aa(A,fun(A,A),times_times(A),D3),C3))
          <=> sums(A,F3,D3) ) ) ) ).

% sums_mult2_iff
tff(fact_2171_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A),A3: A] :
          ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cp(A,fun(fun(nat,A),fun(nat,A)),C3),F3),A3)
         => ( ( C3 != zero_zero(A) )
           => sums(A,F3,divide_divide(A,A3,C3)) ) ) ) ).

% sums_mult_D
tff(fact_2172_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),S2: A] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( sums(A,aTP_Lamp_cq(fun(nat,A),fun(nat,A),F3),S2)
           => sums(A,F3,S2) ) ) ) ).

% sums_Suc_imp
tff(fact_2173_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),S2: A] :
          ( sums(A,aTP_Lamp_cq(fun(nat,A),fun(nat,A),F3),S2)
        <=> sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),S2),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_2174_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_cr(fun(nat,A),fun(nat,A),F3),L)
         => sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_2175_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_cs(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))),N)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_2176_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z2: A,N: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),H)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),N)),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_cu(A,fun(A,fun(nat,fun(nat,A))),H),Z2),N)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_2177_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A3,N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A3,N) = set_fo6178422350223883121st_nat(A,aTP_Lamp_cv(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),one_one(A)) ) ) ) ) ).

% pochhammer_code
tff(fact_2178_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = semiri8178284476397505188at_aux(A,aTP_Lamp_cw(A,A),N,zero_zero(A)) ) ).

% of_nat_code
tff(fact_2179_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),N),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),N)))),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))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_2180_concat__bit__Suc,axiom,
    ! [N: nat,K2: int,L: int] : bit_concat_bit(aa(nat,nat,suc,N),K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K2,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))),bit_concat_bit(N,divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2))),L))) ).

% concat_bit_Suc
tff(fact_2181_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_2182_lessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,set(A),set_ord_lessThan(A),X) = aa(A,set(A),set_ord_lessThan(A),Y) )
        <=> ( X = Y ) ) ) ).

% lessThan_eq_iff
tff(fact_2183_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),aa(A,set(A),set_ord_lessThan(A),K2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),K2)) ) ) ).

% lessThan_iff
tff(fact_2184_finite__lessThan,axiom,
    ! [K2: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(nat,set(nat),set_ord_lessThan(nat),K2))) ).

% finite_lessThan
tff(fact_2185_concat__bit__0,axiom,
    ! [K2: int,L: int] : bit_concat_bit(zero_zero(nat),K2,L) = L ).

% concat_bit_0
tff(fact_2186_card__lessThan,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),aa(nat,set(nat),set_ord_lessThan(nat),U)) = U ).

% card_lessThan
tff(fact_2187_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_2188_prod__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A6) = zero_zero(A) )
          <=> ? [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A6))
                & ( aa(B,A,F3,X4) = zero_zero(A) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_2189_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_lessThan(A),X)),aa(A,set(A),set_ord_lessThan(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% lessThan_subset_iff
tff(fact_2190_lessThan__0,axiom,
    aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_2191_dbl__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K2)) = aa(num,A,numeral_numeral(A),bit0(K2)) ) ).

% dbl_simps(5)
tff(fact_2192_prod_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),X: B,G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A6))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),insert(B,X),A6)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G3,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6)) ) ) ) ) ).

% prod.insert
tff(fact_2193_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,A,G3,N)) ) ).

% sum.lessThan_Suc
tff(fact_2194_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,K2),bot_bot(set(A)))),aa(A,set(A),set_ord_lessThan(A),K2)) = aa(set(A),set(A),insert(A,K2),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_2195_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,A,G3,N)) ) ).

% prod.lessThan_Suc
tff(fact_2196_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M2: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_2197_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_cx(fun(nat,A),fun(nat,fun(nat,A)),G3),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.nat_diff_reindex
tff(fact_2198_lessThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] : aa(A,set(A),set_ord_lessThan(A),X) != bot_bot(set(A)) ) ).

% lessThan_non_empty
tff(fact_2199_infinite__Iio,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [A3: A] : ~ pp(aa(set(A),bool,finite_finite2(A),aa(A,set(A),set_ord_lessThan(A),A3))) ) ).

% infinite_Iio
tff(fact_2200_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),H: fun(B,A),A6: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_cy(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H)),A6) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),A6)) ) ).

% prod.distrib
tff(fact_2201_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_lessThan(A),U) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_cz(A,fun(A,bool),U)) ) ).

% lessThan_def
tff(fact_2202_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_da(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_2203_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_da(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_2204_prod__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(B,A,G3,I3))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6))) ) ) ).

% prod_mono
tff(fact_2205_prod__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A6))) ) ) ).

% prod_nonneg
tff(fact_2206_prod__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A6))) ) ) ).

% prod_pos
tff(fact_2207_prod__ge__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A6))) ) ) ).

% prod_ge_1
tff(fact_2208_prod__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ? [X5: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A6))
                & ( aa(B,A,F3,X5) = zero_zero(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A6) = zero_zero(A) ) ) ) ) ).

% prod_zero
tff(fact_2209_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F3: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F3),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_db(fun(nat,A),fun(nat,fun(A,A)),F3),A3,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_2210_Iio__eq__empty__iff,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & order_bot(A) )
     => ! [N: A] :
          ( ( aa(A,set(A),set_ord_lessThan(A),N) = bot_bot(set(A)) )
        <=> ( N = bot_bot(A) ) ) ) ).

% Iio_eq_empty_iff
tff(fact_2211_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M2: A,N: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(A,set(A),set_ord_lessThan(A),M2)),aa(A,set(A),set_ord_lessThan(A),N)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),N)) ) ) ).

% lessThan_strict_subset_iff
tff(fact_2212_lessThan__Suc,axiom,
    ! [K2: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K2)) = aa(set(nat),set(nat),insert(nat,K2),aa(nat,set(nat),set_ord_lessThan(nat),K2)) ).

% lessThan_Suc
tff(fact_2213_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa(nat,set(nat),set_ord_lessThan(nat),N) = bot_bot(set(nat)) )
    <=> ( N = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_2214_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_2215_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_da(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_2216_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M2: nat,K2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dc(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_2217_prod__le__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),one_one(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A6)),one_one(A))) ) ) ).

% prod_le_1
tff(fact_2218_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R2: fun(A,fun(A,bool)),S3: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R2,one_one(A)),one_one(A)))
         => ( ! [X15: A,Y15: A,X23: A,Y23: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X15),X23))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R2,Y15),Y23)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(A,A,aa(A,fun(A,A),times_times(A),X15),Y15)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y23))) )
           => ( pp(aa(set(B),bool,finite_finite2(B),S3))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(B,A,H,X3)),aa(B,A,G3,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),S3))) ) ) ) ) ) ).

% prod.related
tff(fact_2219_prod_Oinsert__if,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),X: B,G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A6))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),insert(B,X),A6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A6))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),insert(B,X),A6)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G3,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6)) ) ) ) ) ) ).

% prod.insert_if
tff(fact_2220_ivl__disj__int__one_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(4)
tff(fact_2221_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),N: nat,I2: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,N),I2) = semiri8178284476397505188at_aux(A,Inc,N,aa(A,A,Inc,I2)) ) ).

% of_nat_aux.simps(2)
tff(fact_2222_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),I2: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),I2) = I2 ) ).

% of_nat_aux.simps(1)
tff(fact_2223_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,N,M2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_dd(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M2)),set_or1337092689740270186AtMost(nat,N,M2)) ) ).

% prod.atLeastAtMost_rev
tff(fact_2224_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_de(fun(nat,A),fun(nat,fun(nat,A)),G3),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.nat_diff_reindex
tff(fact_2225_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P2: fun(A,nat),N: A] :
          ( ! [X3: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Q,X3)),aa(A,nat,P2,X3)))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P2),aa(A,set(A),set_ord_lessThan(A),N))),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),N))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_df(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P2)),aa(A,set(A),set_ord_lessThan(A),N)) ) ) ) ).

% sum_diff_distrib
tff(fact_2226_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I2: A,F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I5))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
           => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F3,I2)))
             => ( ! [I3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F3,I3))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),I5))) ) ) ) ) ) ).

% less_1_prod2
tff(fact_2227_prod_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B6: set(B),A6: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),A6))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),B6))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),B6)) ) ) ) ) ).

% prod.subset_diff
tff(fact_2228_prod_Ounion__inter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),B6: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B6))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B6))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),B6)) ) ) ) ) ).

% prod.union_inter
tff(fact_2229_prod_OInt__Diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),G3: fun(B,A),B6: set(B)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B6))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),B6))) ) ) ) ).

% prod.Int_Diff
tff(fact_2230_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_2231_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_2232_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_2233_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,K2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K2))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),K2)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K2))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),K2)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ) ) ).

% Iio_Int_singleton
tff(fact_2234_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N))),aa(nat,A,G3,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_da(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_2235_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bk(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_2236_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dg(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_lessThan(nat),M2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,M2)) ) ).

% sum_lessThan_telescope'
tff(fact_2237_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bp(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_lessThan(nat),M2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,M2)),aa(nat,A,F3,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_2238_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F3: fun(nat,A),N: nat,R: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),R)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dh(fun(nat,A),fun(A,fun(nat,A)),F3),R)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sumr_diff_mult_const2
tff(fact_2239_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bk(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_2240_prod__mono__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3)))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,I3)),aa(B,A,G3,I3))) ) )
           => ( ( A6 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6))) ) ) ) ) ).

% prod_mono_strict
tff(fact_2241_prod_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),X: B,G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A6))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G3,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ) ).

% prod.remove
tff(fact_2242_prod_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),G3: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),insert(B,X),A6)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G3,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ).

% prod.insert_remove
tff(fact_2243_prod_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),B6: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B6)))
                 => ( aa(B,A,G3,X3) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),B6)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_2244_prod_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),B6: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B6))
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B6) = bot_bot(set(B)) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),B6)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_2245_binomial__maximum_H,axiom,
    ! [N: nat,K2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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))),N)),K2)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),N))) ).

% binomial_maximum'
tff(fact_2246_binomial__mono,axiom,
    ! [K2: nat,K7: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),K7))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K2)),aa(nat,nat,binomial(N),K7))) ) ) ).

% binomial_mono
tff(fact_2247_binomial__antimono,axiom,
    ! [K2: nat,K7: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),K7))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))),K2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K7),N))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K7)),aa(nat,nat,binomial(N),K2))) ) ) ) ).

% binomial_antimono
tff(fact_2248_binomial__maximum,axiom,
    ! [N: nat,K2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K2)),aa(nat,nat,binomial(N),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% binomial_maximum
tff(fact_2249_prod_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),B6: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B6))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),B6))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B6),A6)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B6))) ) ) ) ) ).

% prod.union_diff2
tff(fact_2250_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A),P: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_2251_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,B2: nat,A3: nat,F3: fun(nat,fun(A,A)),Acc2: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
       => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = Acc2 ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
       => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = set_fo6178422350223883121st_nat(A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F3,A3),Acc2)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_2252_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa2: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa2,Xb,Xc) = Y )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa2))
         => ( Y = Xc ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa2))
         => ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa2),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_2253_prod_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S3: set(B),A3: B,B2: fun(B,A),C3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_di(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C3)),S3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),C3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),aa(set(B),set(B),insert(B,A3),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_di(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C3)),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),C3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),aa(set(B),set(B),insert(B,A3),bot_bot(set(B))))) ) ) ) ) ) ).

% prod.delta_remove
tff(fact_2254_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% power_diff_1_eq
tff(fact_2255_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% one_diff_power_eq
tff(fact_2256_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,N: nat] :
          ( ( X != one_one(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),one_one(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_2257_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B6: set(A),A6: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),B6))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
           => ( ! [B4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B6),A6)))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F3,B4))) )
             => ( ! [A5: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A6))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,A5))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A6)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),B6))) ) ) ) ) ) ).

% prod_mono2
tff(fact_2258_prod__le__power,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(A)
     => ! [A6: set(B),F3: fun(B,A),N: A,K2: nat] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),N)) ) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A6)),K2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A6)),aa(nat,A,aa(A,fun(nat,A),power_power(A),N),K2))) ) ) ) ) ).

% prod_le_power
tff(fact_2259_prod__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [A6: set(B),B6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(B),bool,finite_finite2(B),B6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B6)))
                 => ( aa(B,A,F3,X3) != zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),B6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B6))) ) ) ) ) ) ).

% prod_Un
tff(fact_2260_prod__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom_divide(A)
     => ! [A6: set(B),F3: fun(B,A),A3: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ( aa(B,A,F3,A3) != zero_zero(A) )
           => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),A6))
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),aa(set(B),set(B),insert(B,A3),bot_bot(set(B))))) = divide_divide(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A6),aa(B,A,F3,A3)) ) )
              & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),A6))
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),aa(set(B),set(B),insert(B,A3),bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A6) ) ) ) ) ) ) ).

% prod_diff1
tff(fact_2261_binomial__less__binomial__Suc,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,binomial(N),K2)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K2)))) ) ).

% binomial_less_binomial_Suc
tff(fact_2262_binomial__strict__antimono,axiom,
    ! [K2: nat,K7: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),K7))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K7),N))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,binomial(N),K7)),aa(nat,nat,binomial(N),K2))) ) ) ) ).

% binomial_strict_antimono
tff(fact_2263_binomial__strict__mono,axiom,
    ! [K2: nat,K7: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),K7))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,binomial(N),K2)),aa(nat,nat,binomial(N),K7))) ) ) ).

% binomial_strict_mono
tff(fact_2264_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))) = aa(nat,nat,binomial(N),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% central_binomial_odd
tff(fact_2265_prod__gen__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S3: set(B),A3: B,B2: fun(B,A),C3: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_dj(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C3)),S3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),C3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(B),nat,finite_card(B),S3)),one_one(nat)))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),S3))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_dj(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C3)),S3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C3),aa(set(B),nat,finite_card(B),S3)) ) ) ) ) ) ).

% prod_gen_delta
tff(fact_2266_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_dk(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_2267_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp_strict
tff(fact_2268_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Z2: A,H: A,M2: 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_dl(A,fun(A,fun(nat,fun(nat,A))),Z2),H),M2)),aa(nat,set(nat),set_ord_lessThan(nat),M2)) = 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_dm(A,fun(A,fun(nat,fun(nat,A))),Z2),H),M2)),aa(nat,set(nat),set_ord_lessThan(nat),M2)) ) ).

% lemma_termdiff1
tff(fact_2269_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dn(A,fun(nat,fun(A,fun(nat,A))),X),N),Y)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)))) ) ).

% diff_power_eq_sum
tff(fact_2270_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_do(A,fun(nat,fun(A,fun(nat,A))),X),N),Y)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% power_diff_sumr2
tff(fact_2271_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,F3: fun(nat,A),K5: A,K2: nat] :
          ( ! [P4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P4),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,P4)),K5)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),K5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),K5))) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_2272_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2),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(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_dp(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% prod.in_pairs
tff(fact_2273_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_dq(fun(nat,A),fun(nat,fun(A,A)),F3),A3,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_2274_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dr(A,fun(nat,fun(nat,A)),A3),N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_2275_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ds(A,fun(nat,fun(nat,A)),X),N)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% one_diff_power_eq'
tff(fact_2276_sum__split__even__odd,axiom,
    ! [F3: fun(nat,real),G3: fun(nat,real),N: 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_dt(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F3),G3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) = 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_du(fun(nat,real),fun(nat,real),F3)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_dv(fun(nat,real),fun(nat,real),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ).

% sum_split_even_odd
tff(fact_2277_zero__less__binomial__iff,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N)) ) ).

% zero_less_binomial_iff
tff(fact_2278_choose__two,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% choose_two
tff(fact_2279_binomial__n__0,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2280_binomial__Suc__Suc,axiom,
    ! [N: nat,K2: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(N),K2)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K2))) ).

% binomial_Suc_Suc
tff(fact_2281_binomial__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( aa(nat,nat,binomial(N),K2) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2)) ) ).

% binomial_eq_0_iff
tff(fact_2282_binomial__1,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(nat,nat,suc,zero_zero(nat))) = N ).

% binomial_1
tff(fact_2283_binomial__0__Suc,axiom,
    ! [K2: nat] : aa(nat,nat,binomial(zero_zero(nat)),aa(nat,nat,suc,K2)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_2284_binomial__Suc__n,axiom,
    ! [N: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),N) = aa(nat,nat,suc,N) ).

% binomial_Suc_n
tff(fact_2285_prod__pos__nat__iff,axiom,
    ! [A: $tType,A6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F3),A6)))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X4))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_2286_prod__int__eq,axiom,
    ! [I2: nat,J2: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I2,J2)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_cm(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I2),aa(nat,int,semiring_1_of_nat(int),J2))) ).

% prod_int_eq
tff(fact_2287_prod__int__plus__eq,axiom,
    ! [I2: nat,J2: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_cm(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I2),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J2)))) ).

% prod_int_plus_eq
tff(fact_2288_binomial__eq__0,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2))
     => ( aa(nat,nat,binomial(N),K2) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2289_Suc__times__binomial,axiom,
    ! [K2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K2)) ).

% Suc_times_binomial
tff(fact_2290_Suc__times__binomial__eq,axiom,
    ! [N: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K2))),aa(nat,nat,suc,K2)) ).

% Suc_times_binomial_eq
tff(fact_2291_binomial__symmetric,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
     => ( aa(nat,nat,binomial(N),K2) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)) ) ) ).

% binomial_symmetric
tff(fact_2292_choose__mult__lemma,axiom,
    ! [M2: nat,R: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),R)),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2)),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),R)),K2)),K2)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),R)),M2)) ).

% choose_mult_lemma
tff(fact_2293_binomial__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),R)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R))) ) ).

% binomial_le_pow
tff(fact_2294_zero__less__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K2))) ) ).

% zero_less_binomial
tff(fact_2295_Suc__times__binomial__add,axiom,
    ! [A3: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A3)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2))),aa(nat,nat,suc,A3))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,B2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2))),A3)) ).

% Suc_times_binomial_add
tff(fact_2296_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K2: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K2)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K2)),aa(nat,nat,suc,K2)) ).

% binomial_Suc_Suc_eq_times
tff(fact_2297_choose__mult,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),M2)),aa(nat,nat,binomial(M2),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),K2)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K2))) ) ) ) ).

% choose_mult
tff(fact_2298_binomial__absorb__comp,axiom,
    ! [N: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)),aa(nat,nat,binomial(N),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K2)) ).

% binomial_absorb_comp
tff(fact_2299_binomial__absorption,axiom,
    ! [K2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K2)) ).

% binomial_absorption
tff(fact_2300_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),Mm: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bk(fun(nat,A),fun(nat,A),F3)),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),F3),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_2301_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),N),aa(nat,A,semiring_1_of_nat(A),K2))),K2)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K2)))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2302_binomial__le__pow2,axiom,
    ! [N: nat,K2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% binomial_le_pow2
tff(fact_2303_choose__reduce__nat,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => ( aa(nat,nat,binomial(N),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K2)) ) ) ) ).

% choose_reduce_nat
tff(fact_2304_times__binomial__minus1__eq,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(nat,nat,binomial(N),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2305_binomial__addition__formula,axiom,
    ! [N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,K2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,suc,K2))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K2)) ) ) ).

% binomial_addition_formula
tff(fact_2306_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dw(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).

% choose_odd_sum
tff(fact_2307_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dx(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).

% choose_even_sum
tff(fact_2308_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A6: set(A),K2: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),aa(set(A),nat,finite_card(A),A6)))
       => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_dy(set(A),fun(nat,fun(list(A),bool)),A6),K2))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_bu(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A6)),K2)),one_one(nat)),aa(set(A),nat,finite_card(A),A6))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_2309_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R: A,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dz(A,fun(nat,A),R)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M2)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R),aa(nat,nat,suc,M2))) ) ).

% gchoose_row_sum_weighted
tff(fact_2310_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( accp(nat,vEBT_v4011308405150292612up_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
             => ~ accp(nat,vEBT_v4011308405150292612up_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
               => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
                       => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) )
                      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
                       => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) )
                   => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_2311_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_2312_atMost__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,set(A),set_ord_atMost(A),X) = aa(A,set(A),set_ord_atMost(A),Y) )
        <=> ( X = Y ) ) ) ).

% atMost_eq_iff
tff(fact_2313_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),aa(A,set(A),set_ord_atMost(A),K2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),K2)) ) ) ).

% atMost_iff
tff(fact_2314_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P2: bool,Q: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P2)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
        <=> ( pp(P2)
           => pp(Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_2315_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa(bool,A,zero_neq_one_of_bool(A),fFalse) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_2316_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P2) = zero_zero(A) )
        <=> ~ pp(P2) ) ) ).

% of_bool_eq_0_iff
tff(fact_2317_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P2: bool,Q: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P2)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
        <=> ( ~ pp(P2)
            & pp(Q) ) ) ) ).

% of_bool_less_iff
tff(fact_2318_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P2) = one_one(A) )
        <=> pp(P2) ) ) ).

% of_bool_eq_1_iff
tff(fact_2319_of__bool__eq_I2_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa(bool,A,zero_neq_one_of_bool(A),fTrue) = one_one(A) ) ) ).

% of_bool_eq(2)
tff(fact_2320_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P2: bool] : aa(nat,A,semiring_1_of_nat(A),aa(bool,nat,zero_neq_one_of_bool(nat),P2)) = aa(bool,A,zero_neq_one_of_bool(A),P2) ) ).

% of_nat_of_bool
tff(fact_2321_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fdisj(P2,Q)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(bool,A,zero_neq_one_of_bool(A),P2)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).

% of_bool_or_iff
tff(fact_2322_finite__atMost,axiom,
    ! [K2: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(nat,set(nat),set_ord_atMost(nat),K2))) ).

% finite_atMost
tff(fact_2323_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> pp(P2) ) ) ).

% zero_less_of_bool_iff
tff(fact_2324_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),X)),aa(A,set(A),set_ord_atMost(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% atMost_subset_iff
tff(fact_2325_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P2)),one_one(A)))
        <=> ~ pp(P2) ) ) ).

% of_bool_less_one_iff
tff(fact_2326_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P2: bool] : aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,P2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(bool,A,zero_neq_one_of_bool(A),P2)) ) ).

% of_bool_not_iff
tff(fact_2327_Suc__0__mod__eq,axiom,
    ! [N: nat] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),aa(nat,nat,suc,zero_zero(nat))))) ).

% Suc_0_mod_eq
tff(fact_2328_gbinomial__0_I2_J,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [K2: nat] : aa(nat,B,gbinomial(B,zero_zero(B)),aa(nat,nat,suc,K2)) = zero_zero(B) ) ).

% gbinomial_0(2)
tff(fact_2329_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_2330_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% gbinomial_Suc0
tff(fact_2331_card__atMost,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),aa(nat,set(nat),set_ord_atMost(nat),U)) = aa(nat,nat,suc,U) ).

% card_atMost
tff(fact_2332_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atMost(A),H2)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),H),H2)) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_2333_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ).

% sum.atMost_Suc
tff(fact_2334_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ).

% prod.atMost_Suc
tff(fact_2335_atMost__0,axiom,
    aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_2336_distinct__swap,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( distinct(A,list_update(A,list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),J2)),J2,aa(nat,A,nth(A,Xs),I2)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_2337_sum__mult__of__bool__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A6: set(B),F3: fun(B,A),P2: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_ea(fun(B,A),fun(fun(B,bool),fun(B,A)),F3),P2)),A6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(fun(B,bool),set(B),collect(B),P2))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_2338_sum__of__bool__mult__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A6: set(B),P2: fun(B,bool),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( 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_eb(fun(B,bool),fun(fun(B,A),fun(B,A)),P2),F3)),A6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(fun(B,bool),set(B),collect(B),P2))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_2339_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: bool] : divide_divide(A,aa(bool,A,zero_neq_one_of_bool(A),B2),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_2340_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A6: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(set(list(A)),bool,finite_finite2(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_dy(set(A),fun(nat,fun(list(A),bool)),A6),N)))) ) ).

% finite_lists_distinct_length_eq
tff(fact_2341_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% one_div_2_pow_eq
tff(fact_2342_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] : divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% bits_1_div_exp
tff(fact_2343_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% one_mod_2_pow_eq
tff(fact_2344_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: bool,Q2: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P) = aa(bool,A,zero_neq_one_of_bool(A),Q2) )
        <=> ( pp(P)
          <=> pp(Q2) ) ) ) ).

% of_bool_eq_iff
tff(fact_2345_of__bool__conj,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P2: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fconj(P2,Q)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),P2)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).

% of_bool_conj
tff(fact_2346_not__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [H: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atMost(A),H) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_2347_infinite__Iic,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [A3: A] : ~ pp(aa(set(A),bool,finite_finite2(A),aa(A,set(A),set_ord_atMost(A),A3))) ) ).

% infinite_Iic
tff(fact_2348_not__Iic__eq__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H2: A,L: A,H: A] : aa(A,set(A),set_ord_atMost(A),H2) != set_or1337092689740270186AtMost(A,L,H) ) ).

% not_Iic_eq_Icc
tff(fact_2349_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_atMost(A),U) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ec(A,fun(A,bool),U)) ) ).

% atMost_def
tff(fact_2350_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P2))) ) ).

% zero_less_eq_of_bool
tff(fact_2351_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P2)),one_one(A))) ) ).

% of_bool_less_eq_one
tff(fact_2352_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: bool] :
          ( ( pp(P)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P) = one_one(A) ) )
          & ( ~ pp(P)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P) = zero_zero(A) ) ) ) ) ).

% of_bool_def
tff(fact_2353_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: fun(A,bool),P: bool] :
          ( pp(aa(A,bool,P2,aa(bool,A,zero_neq_one_of_bool(A),P)))
        <=> ( ( pp(P)
             => pp(aa(A,bool,P2,one_one(A))) )
            & ( ~ pp(P)
             => pp(aa(A,bool,P2,zero_zero(A))) ) ) ) ) ).

% split_of_bool
tff(fact_2354_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: fun(A,bool),P: bool] :
          ( pp(aa(A,bool,P2,aa(bool,A,zero_neq_one_of_bool(A),P)))
        <=> ~ ( ( pp(P)
                & ~ pp(aa(A,bool,P2,one_one(A))) )
              | ( ~ pp(P)
                & ~ pp(aa(A,bool,P2,zero_zero(A))) ) ) ) ) ).

% split_of_bool_asm
tff(fact_2355_atMost__atLeast0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),N) = set_or1337092689740270186AtMost(nat,zero_zero(nat),N) ).

% atMost_atLeast0
tff(fact_2356_lessThan__Suc__atMost,axiom,
    ! [K2: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K2)) = aa(nat,set(nat),set_ord_atMost(nat),K2) ).

% lessThan_Suc_atMost
tff(fact_2357_atMost__Suc,axiom,
    ! [K2: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K2)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K2)),aa(nat,set(nat),set_ord_atMost(nat),K2)) ).

% atMost_Suc
tff(fact_2358_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L2: A,H2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),set_or1337092689740270186AtMost(A,L2,H2))) ) ).

% not_Iic_le_Icc
tff(fact_2359_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,J2: nat] :
      ( distinct(A,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
         => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs),J2) )
          <=> ( I2 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_2360_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
         => ! [J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( ( I != J )
               => ( aa(nat,A,nth(A,Xs),I) != aa(nat,A,nth(A,Xs),J) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_2361_card__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) )
     => distinct(A,Xs) ) ).

% card_distinct
tff(fact_2362_distinct__card,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ).

% distinct_card
tff(fact_2363_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A3),K2)),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K2))) ) ).

% gbinomial_Suc_Suc
tff(fact_2364_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K2) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_2365_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),A3)),aa(A,set(A),set_ord_lessThan(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% Iic_subset_Iio_iff
tff(fact_2366_sum__choose__upper,axiom,
    ! [M2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ed(nat,fun(nat,nat),M2)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,M2)) ).

% sum_choose_upper
tff(fact_2367_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ? [X3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs)))
            & ( aa(nat,A,nth(A,Xs),X3) = X )
            & ! [Y4: nat] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y4),aa(list(A),nat,size_size(list(A)),Xs)))
                  & ( aa(nat,A,nth(A,Xs),Y4) = X ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_2368_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M2: nat,A3: 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_ee(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M2)) = 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_ef(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M2)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_2369_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K2)) ) ).

% gbinomial_addition_formula
tff(fact_2370_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K2))),aa(nat,A,gbinomial(A,A3),K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K2)) ) ).

% gbinomial_absorb_comp
tff(fact_2371_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K2: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K2)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,A3,aa(nat,A,semiring_1_of_nat(A),K2))),K2)),aa(nat,A,gbinomial(A,A3),K2))) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_2372_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K2)),aa(nat,A,gbinomial(A,A3),K2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K2)))) ) ).

% gbinomial_mult_1'
tff(fact_2373_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,A3),K2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K2)),aa(nat,A,gbinomial(A,A3),K2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K2)))) ) ).

% gbinomial_mult_1
tff(fact_2374_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bk(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% sum.atMost_Suc_shift
tff(fact_2375_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),I2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dg(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_atMost(nat),I2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,aa(nat,nat,suc,I2))) ) ).

% sum_telescope
tff(fact_2376_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),N: nat,D3: fun(nat,A)] :
          ( ! [X4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_eg(fun(nat,A),fun(A,fun(nat,A)),C3),X4)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_eg(fun(nat,A),fun(A,fun(nat,A)),D3),X4)),aa(nat,set(nat),set_ord_atMost(nat),N))
        <=> ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N))
             => ( aa(nat,A,C3,I) = aa(nat,A,D3,I) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_2377_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_da(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% prod.atMost_Suc_shift
tff(fact_2378_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eh(fun(nat,fun(nat,A)),fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ej(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.nested_swap'
tff(fact_2379_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),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_2380_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),insert(A,U),bot_bot(set(A)))) = aa(A,set(A),set_ord_atMost(A),U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_2381_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ek(fun(nat,fun(nat,A)),fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_em(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.nested_swap'
tff(fact_2382_sum__choose__lower,axiom,
    ! [R: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_en(nat,fun(nat,nat),R)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R),N))),N) ).

% sum_choose_lower
tff(fact_2383_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,gbinomial(A,A3),K2)) ) ).

% Suc_times_gbinomial
tff(fact_2384_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K2)) ) ).

% gbinomial_absorption
tff(fact_2385_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,M2: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),M2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),M2)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M2)),K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K2)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),K2))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_2386_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C3: fun(nat,A),N: nat,K2: nat] :
          ( ! [W: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_eo(fun(nat,A),fun(A,fun(nat,A)),C3),W)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
           => ( aa(nat,A,C3,K2) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_2387_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),N: nat] :
          ( ! [X4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_eg(fun(nat,A),fun(A,fun(nat,A)),C3),X4)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A)
        <=> ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N))
             => ( aa(nat,A,C3,I) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_2388_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bk(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% sum.atMost_shift
tff(fact_2389_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_atMost(nat),M2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)))) ) ).

% sum_up_index_split
tff(fact_2390_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_da(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% prod.atMost_shift
tff(fact_2391_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M2: 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),M2))),one_one(A)))),aa(nat,set(nat),set_ord_atMost(nat),M2)) = aa(nat,A,aa(A,fun(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))),M2)) ) ).

% gbinomial_r_part_sum
tff(fact_2392_sum__choose__diagonal,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_ep(nat,fun(nat,fun(nat,nat)),M2),N)),aa(nat,set(nat),set_ord_atMost(nat),M2)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),M2) ) ) ).

% sum_choose_diagonal
tff(fact_2393_vandermonde,axiom,
    ! [M2: nat,N: nat,R: 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_eq(nat,fun(nat,fun(nat,fun(nat,nat))),M2),N),R)),aa(nat,set(nat),set_ord_atMost(nat),R)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),R) ).

% vandermonde
tff(fact_2394_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P2: fun(A,bool),A3: A] :
          ( ! [A5: A] :
              ( ( divide_divide(A,A5,aa(num,A,numeral_numeral(A),bit0(one2))) = A5 )
             => pp(aa(A,bool,P2,A5)) )
         => ( ! [A5: A,B4: bool] :
                ( pp(aa(A,bool,P2,A5))
               => ( ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,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))),A5)),aa(num,A,numeral_numeral(A),bit0(one2))) = A5 )
                 => pp(aa(A,bool,P2,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,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))),A5)))) ) )
           => pp(aa(A,bool,P2,A3)) ) ) ) ).

% bits_induct
tff(fact_2395_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dz(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M2)),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_2396_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2)))),aa(nat,A,gbinomial(A,A3),K2)) ) ).

% gbinomial_factors
tff(fact_2397_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2)))) ) ).

% gbinomial_rec
tff(fact_2398_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ).

% sum_gp_basic
tff(fact_2399_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(nat,fun(A,bool),aTP_Lamp_er(fun(nat,A),fun(nat,fun(A,bool)),C3),N))))
        <=> ? [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N))
              & ( aa(nat,A,C3,I) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_2400_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K2: nat,N: nat] :
          ( ( aa(nat,A,C3,K2) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
           => pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(nat,fun(A,bool),aTP_Lamp_er(fun(nat,A),fun(nat,fun(A,bool)),C3),N)))) ) ) ) ).

% polyfun_roots_finite
tff(fact_2401_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K2: nat,N: nat] :
          ( ( aa(nat,A,C3,K2) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(nat,fun(A,bool),aTP_Lamp_er(fun(nat,A),fun(nat,fun(A,bool)),C3),N)))),N)) ) ) ) ).

% polyfun_roots_card
tff(fact_2402_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: fun(nat,A),A3: A,N: nat] :
          ( ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),C3),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) )
         => ~ ! [B4: fun(nat,A)] :
                ~ ! [Z4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),C3),Z4)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z4),A3)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),B4),Z4)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).

% polyfun_linear_factor_root
tff(fact_2403_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: fun(nat,A),N: nat,A3: A] :
        ? [B4: fun(nat,A)] :
        ! [Z4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),C3),Z4)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z4),A3)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),B4),Z4)),aa(nat,set(nat),set_ord_lessThan(nat),N)))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),C3),A3)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% polyfun_linear_factor
tff(fact_2404_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M2: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)))) ) ) ) ).

% sum_power_shift
tff(fact_2405_binomial,axiom,
    ! [A3: nat,B2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),N) = 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_et(nat,fun(nat,fun(nat,fun(nat,nat))),A3),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ).

% binomial
tff(fact_2406_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atMost(nat),N)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_2407_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M2: nat,N: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)) ) ).

% exp_mod_exp
tff(fact_2408_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
         => ( aa(nat,A,gbinomial(A,A3),K2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K2)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_2409_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bs(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% sum.in_pairs_0
tff(fact_2410_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K2: nat,N: nat] :
          ( ( aa(nat,A,C3,K2) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
           => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(nat,fun(A,bool),aTP_Lamp_er(fun(nat,A),fun(nat,fun(A,bool)),C3),N))))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(nat,fun(A,bool),aTP_Lamp_er(fun(nat,A),fun(nat,fun(A,bool)),C3),N)))),N)) ) ) ) ) ).

% polyfun_rootbound
tff(fact_2411_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M2: nat,A3: fun(nat,A),N: nat,B2: fun(nat,A),X: A] :
          ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),I3))
             => ( aa(nat,A,A3,I3) = zero_zero(A) ) )
         => ( ! [J3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),J3))
               => ( aa(nat,A,B2,J3) = 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_es(fun(nat,A),fun(A,fun(nat,A)),A3),X)),aa(nat,set(nat),set_ord_atMost(nat),M2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),B2),X)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_ev(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A3),B2),X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) ) ) ) ) ).

% polynomial_product
tff(fact_2412_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_dp(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% prod.in_pairs_0
tff(fact_2413_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),N: nat,K2: A] :
          ( ! [X4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_eg(fun(nat,A),fun(A,fun(nat,A)),C3),X4)),aa(nat,set(nat),set_ord_atMost(nat),N)) = K2
        <=> ( ( aa(nat,A,C3,zero_zero(nat)) = K2 )
            & ! [X4: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or1337092689740270186AtMost(nat,one_one(nat),N)))
               => ( aa(nat,A,C3,X4) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_2414_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),N) = 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_ew(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% binomial_ring
tff(fact_2415_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,B2: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),N) = 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_ex(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% pochhammer_binomial_sum
tff(fact_2416_polynomial__product__nat,axiom,
    ! [M2: nat,A3: fun(nat,nat),N: nat,B2: fun(nat,nat),X: nat] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),I3))
         => ( aa(nat,nat,A3,I3) = zero_zero(nat) ) )
     => ( ! [J3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),J3))
           => ( aa(nat,nat,B2,J3) = 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_ey(fun(nat,nat),fun(nat,fun(nat,nat)),A3),X)),aa(nat,set(nat),set_ord_atMost(nat),M2))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_ey(fun(nat,nat),fun(nat,fun(nat,nat)),B2),X)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_fa(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A3),B2),X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_2417_choose__square__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fb(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),N) ).

% choose_square_sum
tff(fact_2418_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),N: nat,X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xs,N,X)) = aa(set(A),set(A),insert(A,X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,aa(nat,A,nth(A,Xs),N)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_2419_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P: nat,K2: nat,G3: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),P))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),P)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_2420_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P: nat,K2: nat,G3: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),P))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fe(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),P)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ff(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_2421_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fg(nat,fun(nat,A),K2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_2422_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M2: nat,N: nat] : divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2)),zero_zero(A))),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N))) ) ).

% exp_div_exp_eq
tff(fact_2423_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
         => ( aa(nat,A,gbinomial(A,A3),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,aa(nat,A,semiring_1_of_nat(A),K2))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_2424_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp0
tff(fact_2425_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A3: fun(nat,A),X: A,Y: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),A3),X)),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),A3),Y)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_fi(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A3),X),Y)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ) ).

% polyfun_diff_alt
tff(fact_2426_binomial__r__part__sum,axiom,
    ! [M2: 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))),M2)),one_one(nat)))),aa(nat,set(nat),set_ord_atMost(nat),M2)) = aa(nat,nat,aa(nat,fun(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))),M2)) ).

% binomial_r_part_sum
tff(fact_2427_choose__linear__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fj(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ).

% choose_linear_sum
tff(fact_2428_card__lists__length__le,axiom,
    ! [A: $tType,A6: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_ak(set(A),fun(nat,fun(list(A),bool)),A6),N))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A6))),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% card_lists_length_le
tff(fact_2429_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E3: real,C3: fun(nat,A),N: nat] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
         => ? [M8: real] :
            ! [Z4: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),M8),real_V7770717601297561774m_norm(A,Z4)))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fk(fun(nat,A),fun(A,fun(nat,A)),C3),Z4)),aa(nat,set(nat),set_ord_atMost(nat),N)))),aa(real,real,aa(real,fun(real,real),times_times(real),E3),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z4)),aa(nat,nat,suc,N))))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_2430_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A3: fun(nat,A),X: A,Y: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),A3),X)),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),A3),Y)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_fm(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A3),X),Y)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ) ).

% polyfun_diff
tff(fact_2431_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K2: nat,A6: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(set(A),nat,finite_card(A),A6)))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),bool),aTP_Lamp_fn(nat,fun(set(A),fun(list(A),bool)),K2),A6))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_bu(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A6)),K2)),one_one(nat)),aa(set(A),nat,finite_card(A),A6))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_2432_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q2),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int))))) ).

% Divides.adjust_div_eq
tff(fact_2433_list__decode_Opinduct,axiom,
    ! [A0: nat,P2: fun(nat,bool)] :
      ( accp(nat,nat_list_decode_rel,A0)
     => ( ( accp(nat,nat_list_decode_rel,zero_zero(nat))
         => pp(aa(nat,bool,P2,zero_zero(nat))) )
       => ( ! [N3: nat] :
              ( accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N3))
             => ( ! [X5: nat,Y4: nat] :
                    ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X5),Y4) = aa(nat,product_prod(nat,nat),nat_prod_decode,N3) )
                   => pp(aa(nat,bool,P2,Y4)) )
               => pp(aa(nat,bool,P2,aa(nat,nat,suc,N3))) ) )
         => pp(aa(nat,bool,P2,A0)) ) ) ) ).

% list_decode.pinduct
tff(fact_2434_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A3: A] :
          ( ( ( K2 = zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A3),K2) = one_one(A) ) )
          & ( ( K2 != zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A3),K2) = divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_fo(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)),one_one(A)),semiring_char_0_fact(A,K2)) ) ) ) ) ).

% gbinomial_code
tff(fact_2435_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fp(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_2436_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_2437_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),nat_set_decode(X)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X)) ) ).

% set_decode_0
tff(fact_2438_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A3)) = A3 ) ).

% add.inverse_inverse
tff(fact_2439_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B2) )
        <=> ( A3 = B2 ) ) ) ).

% neg_equal_iff_equal
tff(fact_2440_of__nat__id,axiom,
    ! [N: nat] : aa(nat,nat,semiring_1_of_nat(nat),N) = N ).

% of_nat_id
tff(fact_2441_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% compl_le_compl_iff
tff(fact_2442_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% neg_le_iff_le
tff(fact_2443_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = A3 )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_2444_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_2445_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_2446_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A3) )
        <=> ( zero_zero(A) = A3 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_2447_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_2448_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% neg_less_iff_less
tff(fact_2449_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M2: num,N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( M2 = N ) ) ) ).

% neg_numeral_eq_iff
tff(fact_2450_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% mult_minus_right
tff(fact_2451_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) ) ).

% minus_mult_minus
tff(fact_2452_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% mult_minus_left
tff(fact_2453_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2)) = B2 ) ).

% add_minus_cancel
tff(fact_2454_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = B2 ) ).

% minus_add_cancel
tff(fact_2455_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_add_distrib
tff(fact_2456_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) ) ).

% minus_diff_eq
tff(fact_2457_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(A,A,uminus_uminus(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y)) ) ) ).

% dvd_minus_iff
tff(fact_2458_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,uminus_uminus(A),X)),Y))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y)) ) ) ).

% minus_dvd_iff
tff(fact_2459_negative__eq__positive,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),M2) )
    <=> ( ( N = zero_zero(nat) )
        & ( M2 = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_2460_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_2461_set__decode__inverse,axiom,
    ! [N: nat] : aa(set(nat),nat,nat_set_encode,nat_set_decode(N)) = N ).

% set_decode_inverse
tff(fact_2462_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% neg_less_eq_nonneg
tff(fact_2463_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% less_eq_neg_nonpos
tff(fact_2464_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% neg_le_0_iff_le
tff(fact_2465_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% neg_0_le_iff_le
tff(fact_2466_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% less_neg_neg
tff(fact_2467_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% neg_less_pos
tff(fact_2468_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% neg_0_less_iff_less
tff(fact_2469_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% neg_less_0_iff_less
tff(fact_2470_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),A3)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_2471_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_2472_verit__minus__simplify_I3_J,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),zero_zero(B)),B2) = aa(B,B,uminus_uminus(B),B2) ) ).

% verit_minus_simplify(3)
tff(fact_2473_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A3) = aa(A,A,uminus_uminus(A),A3) ) ).

% diff_0
tff(fact_2474_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_2475_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_2476_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_2477_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) ) ).

% diff_minus_eq_add
tff(fact_2478_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) ) ).

% uminus_add_conv_diff
tff(fact_2479_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_2480_negative__zless,axiom,
    ! [N: nat,M2: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),aa(nat,int,semiring_1_of_nat(int),M2))) ).

% negative_zless
tff(fact_2481_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_2482_dbl__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K2))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K2))) ) ).

% dbl_simps(1)
tff(fact_2483_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_2484_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_2485_set__encode__inverse,axiom,
    ! [A6: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A6)) = A6 ) ) ).

% set_encode_inverse
tff(fact_2486_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_2487_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_2488_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% diff_numeral_special(12)
tff(fact_2489_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( N = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2490_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2491_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),A3)) = A3 ) ).

% left_minus_one_mult_self
tff(fact_2492_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_2493_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A] : modulo_modulo(A,A3,aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_2494_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))))
           => ( 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),V3))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))))
           => ( 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),V3))) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(2)
tff(fact_2495_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V3)))
           => ( 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),V3)) = aa(num,A,numeral_numeral(A),V3) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V3)))
           => ( 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),V3)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(3)
tff(fact_2496_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))))
           => ( 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),V3))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))))
           => ( 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),V3))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(4)
tff(fact_2497_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_2498_fact__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N))),semiring_char_0_fact(A,N)) ) ).

% fact_Suc
tff(fact_2499_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N))) ) ).

% diff_numeral_simps(3)
tff(fact_2500_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)) ) ).

% diff_numeral_simps(2)
tff(fact_2501_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_2502_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_2503_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_2504_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V3: 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),V3))),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),V3),W2))),Y) ) ).

% semiring_norm(172)
tff(fact_2505_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V3: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V3)),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),V3),W2)))),Y) ) ).

% semiring_norm(171)
tff(fact_2506_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V3: 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),V3))),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),V3),W2)))),Y) ) ).

% semiring_norm(170)
tff(fact_2507_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M2)) ) ) ).

% neg_numeral_le_iff
tff(fact_2508_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M2)) ) ) ).

% neg_numeral_less_iff
tff(fact_2509_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_2510_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))))
        <=> ( M2 != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2511_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2)) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_2512_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_2513_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W2: num,A3: A] :
          ( ( divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = A3 )
        <=> ( ( ( 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),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_2514_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,W2: num] :
          ( ( A3 = divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) )
        <=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = B2 ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_2515_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),one_one(A))))
        <=> ( M2 != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2516_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_2517_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2)) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_2518_dbl__dec__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K2))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K2))) ) ).

% dbl_dec_simps(1)
tff(fact_2519_dbl__inc__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K2))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K2))) ) ).

% dbl_inc_simps(1)
tff(fact_2520_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K2: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K2)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_2521_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K2: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),bit0(K2))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K2))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_bit0
tff(fact_2522_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_2523_diff__numeral__special_I11_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_2524_diff__numeral__special_I10_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_2525_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_2526_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),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),M2),one2))) ) ).

% diff_numeral_special(4)
tff(fact_2527_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% diff_numeral_special(3)
tff(fact_2528_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,N)),nat_set_decode(X)))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),nat_set_decode(divide_divide(nat,X,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% set_decode_Suc
tff(fact_2529_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_2530_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = one_one(A) ) ).

% power_minus1_even
tff(fact_2531_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),A3) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_2532_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% equation_minus_iff
tff(fact_2533_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B2 )
        <=> ( aa(A,A,uminus_uminus(A),B2) = A3 ) ) ) ).

% minus_equation_iff
tff(fact_2534_fact__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),semiring_char_0_fact(nat,M2)),semiring_char_0_fact(nat,N))) ) ).

% fact_mono_nat
tff(fact_2535_fact__ge__self,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),semiring_char_0_fact(nat,N))) ).

% fact_ge_self
tff(fact_2536_fact__nonzero,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semiri3467727345109120633visors(A) )
     => ! [N: nat] : semiring_char_0_fact(A,N) != zero_zero(A) ) ).

% fact_nonzero
tff(fact_2537_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% compl_le_swap2
tff(fact_2538_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% compl_le_swap1
tff(fact_2539_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X))) ) ) ).

% compl_mono
tff(fact_2540_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% le_imp_neg_le
tff(fact_2541_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A3)) ) ) ).

% minus_le_iff
tff(fact_2542_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% le_minus_iff
tff(fact_2543_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,uminus_uminus(A),A3))) ) ) ).

% less_minus_iff
tff(fact_2544_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A3)) ) ) ).

% minus_less_iff
tff(fact_2545_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M2: num,N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)) != aa(num,A,numeral_numeral(A),N) ) ).

% neg_numeral_neq_numeral
tff(fact_2546_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M2: num,N: num] : aa(num,A,numeral_numeral(A),M2) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_neq_neg_numeral
tff(fact_2547_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_mult_commute
tff(fact_2548_square__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),B2) )
        <=> ( ( A3 = B2 )
            | ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% square_eq_iff
tff(fact_2549_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),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),A3)) ) ).

% is_num_normalize(8)
tff(fact_2550_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A6: A,K2: A,A3: A] :
          ( ( A6 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),A3) )
         => ( aa(A,A,uminus_uminus(A),A6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K2)),aa(A,A,uminus_uminus(A),A3)) ) ) ) ).

% group_cancel.neg1
tff(fact_2551_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),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),A3)) ) ).

% add.inverse_distrib_swap
tff(fact_2552_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_2553_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),B2)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).

% minus_diff_commute
tff(fact_2554_signed__take__bit__mult,axiom,
    ! [N: nat,K2: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),K2),L)) ).

% signed_take_bit_mult
tff(fact_2555_fact__less__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),semiring_char_0_fact(nat,M2)),semiring_char_0_fact(nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_2556_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_zero
tff(fact_2557_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,N)),zero_zero(A))) ) ).

% fact_not_neg
tff(fact_2558_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_gt_zero
tff(fact_2559_finite__set__decode,axiom,
    ! [N: nat] : pp(aa(set(nat),bool,finite_finite2(nat),nat_set_decode(N))) ).

% finite_set_decode
tff(fact_2560_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_1
tff(fact_2561_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_le_numeral
tff(fact_2562_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_le_neg_numeral
tff(fact_2563_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% zero_neq_neg_numeral
tff(fact_2564_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_less_neg_numeral
tff(fact_2565_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num,N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_less_numeral
tff(fact_2566_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(4)
tff(fact_2567_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% le_minus_one_simps(2)
tff(fact_2568_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% add_eq_0_iff
tff(fact_2569_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_2570_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A3) = B2 ) ) ) ).

% add.inverse_unique
tff(fact_2571_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2572_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2573_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_2574_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(4)
tff(fact_2575_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% less_minus_one_simps(2)
tff(fact_2576_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_2577_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,uminus_uminus(A),A3),aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,A3,B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2578_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),divide_divide(A,A3,B2)) = divide_divide(A,A3,aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_2579_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% one_neq_neg_numeral
tff(fact_2580_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),N) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% numeral_neq_neg_one
tff(fact_2581_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_2582_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,M2)),semiring_char_0_fact(A,N))) ) ) ).

% fact_mono
tff(fact_2583_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2584_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% diff_conv_add_uminus
tff(fact_2585_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B6: A,K2: A,B2: A,A3: A] :
          ( ( B6 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),B2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_2586_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( divide_divide(A,aa(A,A,uminus_uminus(A),A3),B2) = aa(A,A,uminus_uminus(A),divide_divide(A,A3,B2)) ) ) ) ).

% dvd_neg_div
tff(fact_2587_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( divide_divide(A,A3,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),divide_divide(A,A3,B2)) ) ) ) ).

% dvd_div_neg
tff(fact_2588_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,M2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,M2))) ) ) ).

% fact_dvd
tff(fact_2589_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : pp(aa(real,bool,aa(real,fun(real,bool),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_2590_int__of__nat__induct,axiom,
    ! [P2: fun(int,bool),Z2: int] :
      ( ! [N3: nat] : pp(aa(int,bool,P2,aa(nat,int,semiring_1_of_nat(int),N3)))
     => ( ! [N3: nat] : pp(aa(int,bool,P2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N3)))))
       => pp(aa(int,bool,P2,Z2)) ) ) ).

% int_of_nat_induct
tff(fact_2591_int__cases,axiom,
    ! [Z2: int] :
      ( ! [N3: nat] : Z2 != aa(nat,int,semiring_1_of_nat(int),N3)
     => ~ ! [N3: nat] : Z2 != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N3))) ) ).

% int_cases
tff(fact_2592_zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M2),N) = one_one(int) )
    <=> ( ( ( M2 = one_one(int) )
          & ( N = one_one(int) ) )
        | ( ( M2 = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_2593_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M2: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M2),N) = one_one(int) )
     => ( ( M2 = one_one(int) )
        | ( M2 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_2594_zmod__zminus1__not__zero,axiom,
    ! [K2: int,L: int] :
      ( ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K2),L) != zero_zero(int) )
     => ( modulo_modulo(int,K2,L) != zero_zero(int) ) ) ).

% zmod_zminus1_not_zero
tff(fact_2595_zmod__zminus2__not__zero,axiom,
    ! [K2: int,L: int] :
      ( ( modulo_modulo(int,K2,aa(int,int,uminus_uminus(int),L)) != zero_zero(int) )
     => ( modulo_modulo(int,K2,L) != zero_zero(int) ) ) ).

% zmod_zminus2_not_zero
tff(fact_2596_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),semiring_char_0_fact(A,N)) ) ).

% pochhammer_same
tff(fact_2597_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_2598_dvd__fact,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M2),semiring_char_0_fact(nat,N))) ) ) ).

% dvd_fact
tff(fact_2599_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_le_neg_numeral
tff(fact_2600_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_le_zero
tff(fact_2601_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_less_neg_numeral
tff(fact_2602_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_less_zero
tff(fact_2603_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% le_minus_one_simps(1)
tff(fact_2604_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(3)
tff(fact_2605_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% less_minus_one_simps(1)
tff(fact_2606_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(3)
tff(fact_2607_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)))) ) ).

% not_one_le_neg_numeral
tff(fact_2608_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_le_neg_one
tff(fact_2609_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% neg_numeral_le_neg_one
tff(fact_2610_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M2))) ) ).

% neg_one_le_numeral
tff(fact_2611_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),one_one(A))) ) ).

% neg_numeral_le_one
tff(fact_2612_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,A3),K2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K2)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A3),K2)),semiring_char_0_fact(A,K2)) ) ).

% gbinomial_pochhammer
tff(fact_2613_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),one_one(A))) ) ).

% neg_numeral_less_one
tff(fact_2614_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M2))) ) ).

% neg_one_less_numeral
tff(fact_2615_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_less_neg_one
tff(fact_2616_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)))) ) ).

% not_one_less_neg_numeral
tff(fact_2617_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_2618_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)) )
        <=> ( ( ( C3 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = aa(A,A,uminus_uminus(A),B2) ) )
            & ( ( C3 = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_2619_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,A3: A] :
          ( ( aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)) = A3 )
        <=> ( ( ( C3 != zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) ) )
            & ( ( C3 = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_2620_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A,C3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),divide_divide(A,A3,B2)) = C3 )
          <=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2621_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C3 = aa(A,A,uminus_uminus(A),divide_divide(A,A3,B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2622_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_2623_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_2624_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( divide_divide(A,A3,B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_2625_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_2626_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,M2)),semiring_char_0_fact(A,N))) ) ) ) ).

% fact_less_mono
tff(fact_2627_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_minus
tff(fact_2628_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% inf_shunt
tff(fact_2629_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: nat,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),semiring_char_0_fact(A,N))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N)))) ) ).

% fact_fact_dvd_fact
tff(fact_2630_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( modulo_modulo(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,M2)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_2631_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P: A,Q2: A,R: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q2),R)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P),aa(A,A,uminus_uminus(A),Q2))),R)) ) ) ).

% sup_neg_inf
tff(fact_2632_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y))),Z2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2))) ) ) ).

% shunt2
tff(fact_2633_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z2))) ) ) ).

% shunt1
tff(fact_2634_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,N)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),N)))) ) ).

% fact_le_power
tff(fact_2635_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N3: nat] : M2 != aa(nat,int,semiring_1_of_nat(int),N3)
     => ~ ! [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
           => ( M2 != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) ) ) ) ).

% int_cases4
tff(fact_2636_int__zle__neg,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M2))))
    <=> ( ( N = zero_zero(nat) )
        & ( M2 = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_2637_zmod__zminus1__eq__if,axiom,
    ! [A3: int,B2: int] :
      ( ( ( modulo_modulo(int,A3,B2) = zero_zero(int) )
       => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),A3),B2) = zero_zero(int) ) )
      & ( ( modulo_modulo(int,A3,B2) != zero_zero(int) )
       => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),A3),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),modulo_modulo(int,A3,B2)) ) ) ) ).

% zmod_zminus1_eq_if
tff(fact_2638_zmod__zminus2__eq__if,axiom,
    ! [A3: int,B2: int] :
      ( ( ( modulo_modulo(int,A3,B2) = zero_zero(int) )
       => ( modulo_modulo(int,A3,aa(int,int,uminus_uminus(int),B2)) = zero_zero(int) ) )
      & ( ( modulo_modulo(int,A3,B2) != zero_zero(int) )
       => ( modulo_modulo(int,A3,aa(int,int,uminus_uminus(int),B2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),modulo_modulo(int,A3,B2)),B2) ) ) ) ).

% zmod_zminus2_eq_if
tff(fact_2639_fact__div__fact__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,semiring_char_0_fact(nat,N),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),R)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R))) ) ).

% fact_div_fact_le_pow
tff(fact_2640_binomial__fact__lemma,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K2)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))),aa(nat,nat,binomial(N),K2)) = semiring_char_0_fact(nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_2641_subset__decode__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),nat_set_decode(M2)),nat_set_decode(N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% subset_decode_imp_le
tff(fact_2642_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_2643_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_2644_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2645_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2646_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2647_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2648_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C3: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = divide_divide(A,B2,C3) )
        <=> ( ( ( C3 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3) = B2 ) )
            & ( ( C3 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2649_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,W2: num] :
          ( ( divide_divide(A,B2,C3) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) )
        <=> ( ( ( C3 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3) ) )
            & ( ( C3 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2650_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,A3: A,B2: A] :
          ( ( ( Z2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z2))),B2) = B2 ) )
          & ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z2))),B2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2)),Z2) ) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_2651_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),divide_divide(A,X,Z2))),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)),Z2) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2652_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,N)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2653_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,A3: A,B2: A] :
          ( ( ( Z2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z2))),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z2))),B2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2)),Z2) ) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2654_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z2: A,A3: A,B2: A] :
          ( ( ( Z2 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,Z2)),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,Z2)),B2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2)),Z2) ) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2655_minus__divide__diff__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),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,X,Z2))),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)),Z2) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2656_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))),semiring_char_0_fact(A,N))) ) ) ).

% choose_dvd
tff(fact_2657_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero(int) )
     => ( ! [N3: nat] :
            ( ( K2 = aa(nat,int,semiring_1_of_nat(int),N3) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3)) )
       => ~ ! [N3: nat] :
              ( ( K2 = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) )
             => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3)) ) ) ) ).

% int_cases3
tff(fact_2658_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,K2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K2) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_2659_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [N: nat,K2: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K2) = zero_zero(A) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2)) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2660_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K3),N))
              & ( A3 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2661_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K2) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_2662_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),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,N))))) ).

% not_zle_0_negative
tff(fact_2663_negD,axiom,
    ! [X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),zero_zero(int)))
     => ? [N3: nat] : X = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N3))) ) ).

% negD
tff(fact_2664_negative__zless__0,axiom,
    ! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),zero_zero(int))) ).

% negative_zless_0
tff(fact_2665_div__eq__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( divide_divide(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_2666_prod_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),P2: fun(B,bool),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_fq(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P2),H),G3)),A6) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(fun(B,bool),set(B),collect(B),P2)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),aa(set(B),set(B),uminus_uminus(set(B)),aa(fun(B,bool),set(B),collect(B),P2))))) ) ) ) ).

% prod.If_cases
tff(fact_2667_binomial__altdef__nat,axiom,
    ! [K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
     => ( aa(nat,nat,binomial(N),K2) = divide_divide(nat,semiring_char_0_fact(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K2)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))) ) ) ).

% binomial_altdef_nat
tff(fact_2668_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_2669_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A3))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_2670_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2671_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2672_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2673_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2674_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2675_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),divide_divide(A,B2,C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_2676_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2677_neg__int__cases,axiom,
    ! [K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
     => ~ ! [N3: nat] :
            ( ( K2 = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3)) ) ) ).

% neg_int_cases
tff(fact_2678_minus__mod__int__eq,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L))
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K2),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),one_one(int))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),one_one(int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_2679_zmod__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_2680_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( ( modulo_modulo(int,A3,B2) = zero_zero(int) )
         => ( divide_divide(int,aa(int,int,uminus_uminus(int),A3),B2) = aa(int,int,uminus_uminus(int),divide_divide(int,A3,B2)) ) )
        & ( ( modulo_modulo(int,A3,B2) != zero_zero(int) )
         => ( divide_divide(int,aa(int,int,uminus_uminus(int),A3),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),divide_divide(int,A3,B2))),one_one(int)) ) ) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_2681_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( ( modulo_modulo(int,A3,B2) = zero_zero(int) )
         => ( divide_divide(int,A3,aa(int,int,uminus_uminus(int),B2)) = aa(int,int,uminus_uminus(int),divide_divide(int,A3,B2)) ) )
        & ( ( modulo_modulo(int,A3,B2) != zero_zero(int) )
         => ( divide_divide(int,A3,aa(int,int,uminus_uminus(int),B2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),divide_divide(int,A3,B2))),one_one(int)) ) ) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_2682_fact__eq__fact__times,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( semiring_char_0_fact(nat,M2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_bu(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M2))) ) ) ).

% fact_eq_fact_times
tff(fact_2683_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ) ) ) ).

% signed_take_bit_rec
tff(fact_2684_square__fact__le__2__fact,axiom,
    ! [N: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),semiring_char_0_fact(real,N)),semiring_char_0_fact(real,N))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ).

% square_fact_le_2_fact
tff(fact_2685_zminus1__lemma,axiom,
    ! [A3: int,B2: int,Q2: int,R: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A3),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int)),aa(int,int,uminus_uminus(int),Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q2)),one_one(int)))),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int)),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R)))) ) ) ).

% zminus1_lemma
tff(fact_2686_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2687_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),divide_divide(A,B2,C3)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_2688_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))) ) ) ) ).

% square_le_1
tff(fact_2689_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% minus_power_mult_self
tff(fact_2690_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M2: nat] :
          ( ( ( M2 = zero_zero(nat) )
           => ( semiring_char_0_fact(A,M2) = one_one(A) ) )
          & ( ( M2 != zero_zero(nat) )
           => ( semiring_char_0_fact(A,M2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat)))) ) ) ) ) ).

% fact_num_eq_if
tff(fact_2691_fact__code,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)),N,one_one(nat))) ) ).

% fact_code
tff(fact_2692_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_2693_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),R),aa(nat,A,semiring_1_of_nat(A),K2))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R),K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),R),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R)),one_one(A)),K2)) ) ).

% pochhammer_absorb_comp
tff(fact_2694_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K2)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K2))),one_one(A))),N)) ) ).

% gbinomial_index_swap
tff(fact_2695_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,A3),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K2)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),K2)),A3)),one_one(A))),K2)) ) ).

% gbinomial_negated_upper
tff(fact_2696_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K2)) = divide_divide(A,semiring_char_0_fact(A,N),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)))) ) ) ) ).

% binomial_fact
tff(fact_2697_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K2))) = divide_divide(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2))) ) ) ) ).

% fact_binomial
tff(fact_2698_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_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),N)),X))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),N))) ) ).

% Bernoulli_inequality
tff(fact_2699_fact__div__fact,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( divide_divide(nat,semiring_char_0_fact(nat,M2),semiring_char_0_fact(nat,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_bu(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M2)) ) ) ).

% fact_div_fact
tff(fact_2700_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(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))),N))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% power_minus1_odd
tff(fact_2701_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A3)),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K2)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),K2))),one_one(A))),K2)) ) ).

% gbinomial_minus
tff(fact_2702_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_2703_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K2))),one_one(A)),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K2)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K2)) ) ).

% pochhammer_minus'
tff(fact_2704_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K2)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K2))),one_one(A)),K2)) ) ).

% pochhammer_minus
tff(fact_2705_int__bit__induct,axiom,
    ! [P2: fun(int,bool),K2: int] :
      ( pp(aa(int,bool,P2,zero_zero(int)))
     => ( pp(aa(int,bool,P2,aa(int,int,uminus_uminus(int),one_one(int))))
       => ( ! [K: int] :
              ( pp(aa(int,bool,P2,K))
             => ( ( K != zero_zero(int) )
               => pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))) ) )
         => ( ! [K: int] :
                ( pp(aa(int,bool,P2,K))
               => ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => pp(aa(int,bool,P2,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),K),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) )
           => pp(aa(int,bool,P2,K2)) ) ) ) ) ).

% int_bit_induct
tff(fact_2706_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fr(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),M2)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),M2)) ) ).

% gbinomial_sum_lower_neg
tff(fact_2707_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K2)) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fs(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K2)),semiring_char_0_fact(A,aa(nat,nat,suc,K2))) ) ).

% gbinomial_Suc
tff(fact_2708_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M2: nat,A3: 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_ee(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M2)) = 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_ft(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M2),A3),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M2)) ) ).

% gbinomial_partial_sum_poly
tff(fact_2709_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,Z2: A,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),N) = A3 )
          <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_fu(nat,fun(A,fun(A,fun(nat,A))),N),Z2),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_2710_set__decode__plus__power__2,axiom,
    ! [N: nat,Z2: nat] :
      ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),nat_set_decode(Z2)))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),Z2)) = aa(set(nat),set(nat),insert(nat,N),nat_set_decode(Z2)) ) ) ).

% set_decode_plus_power_2
tff(fact_2711_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( ( N != one_one(nat) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fv(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_2712_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_fw(nat,fun(nat,bool),X)) ).

% set_decode_def
tff(fact_2713_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N: 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))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(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))),N))),comm_s3205402744901411588hammer(A,divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))),semiring_char_0_fact(A,N)) ) ).

% fact_double
tff(fact_2714_binomial__code,axiom,
    ! [N: nat,K2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2))
       => ( aa(nat,nat,binomial(N),K2) = zero_zero(nat) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)))
           => ( aa(nat,nat,binomial(N),K2) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)))
           => ( aa(nat,nat,binomial(N),K2) = divide_divide(nat,set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2)),one_one(nat)),N,one_one(nat)),semiring_char_0_fact(nat,K2)) ) ) ) ) ) ).

% binomial_code
tff(fact_2715_Maclaurin__lemma,axiom,
    ! [H: real,F3: fun(real,real),J2: fun(nat,real),N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ? [B9: real] : aa(real,real,F3,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,real),fun(nat,real),aTP_Lamp_fx(real,fun(fun(nat,real),fun(nat,real)),H),J2)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),B9),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N),semiring_char_0_fact(real,N)))) ) ).

% Maclaurin_lemma
tff(fact_2716_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: real,N: nat,Diff: fun(nat,fun(A,real))] :
          ( ( X = zero_zero(real) )
         => ( ( N != zero_zero(nat) )
           => ( aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_fy(real,fun(fun(nat,fun(A,real)),fun(nat,real)),X),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_2717_sin__coeff__def,axiom,
    ! [X5: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X5))
       => ( sin_coeff(X5) = zero_zero(real) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X5))
       => ( sin_coeff(X5) = divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X5),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),semiring_char_0_fact(real,X5)) ) ) ) ).

% sin_coeff_def
tff(fact_2718_fact__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M2)))
     => ( semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M2)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M2)),N)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N))) ) ) ).

% fact_diff_Suc
tff(fact_2719_sin__paired,axiom,
    ! [X: real] : sums(real,aTP_Lamp_fz(real,fun(nat,real),X),sin(real,X)) ).

% sin_paired
tff(fact_2720_and__int_Osimps,axiom,
    ! [K2: int,L: int] :
      ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
          & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K2)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))) ) )
      & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K2)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.simps
tff(fact_2721_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_2722_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_2723_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_2724_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_2725_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_2726_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_2727_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_2728_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_2729_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_2730_and__int__rec,axiom,
    ! [K2: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K2)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% and_int_rec
tff(fact_2731_and__int__unfold,axiom,
    ! [K2: int,L: int] :
      ( ( ( ( K2 = zero_zero(int) )
          | ( L = zero_zero(int) ) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = zero_zero(int) ) )
      & ( ~ ( ( K2 = zero_zero(int) )
            | ( L = zero_zero(int) ) )
       => ( ( ( K2 = aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = L ) )
          & ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = K2 ) )
              & ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = 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,K2,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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_2732_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa2) = Y )
     => ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa2))))) ) )
        & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
              & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,X,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,Xa2,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ).

% and_int.elims
tff(fact_2733_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ? [T6: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T6))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T6),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_ga(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_2734_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ? [T6: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T6))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T6),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_ga(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_2735_sumr__cos__zero__one,axiom,
    ! [N: nat] : aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_gb(nat,real)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_2736_sin__cos__npi,axiom,
    ! [N: nat] : sin(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))),N)))),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% sin_cos_npi
tff(fact_2737_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T6: 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_ga(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ).

% Maclaurin_sin_expansion
tff(fact_2738_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K2)))) = 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,N),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2))),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_2739_semiring__norm_I15_J,axiom,
    ! [M2: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M2)),bit0(N)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M2)),N)) ).

% semiring_norm(15)
tff(fact_2740_semiring__norm_I14_J,axiom,
    ! [M2: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M2)),aa(num,num,bit1,N)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M2),aa(num,num,bit1,N))) ).

% semiring_norm(14)
tff(fact_2741_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_2742_dbl__inc__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K2)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,K2)) ) ).

% dbl_inc_simps(5)
tff(fact_2743_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_2744_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_2745_zdiv__numeral__Bit1,axiom,
    ! [V3: num,W2: num] : divide_divide(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,V3)),aa(num,int,numeral_numeral(int),bit0(W2))) = divide_divide(int,aa(num,int,numeral_numeral(int),V3),aa(num,int,numeral_numeral(int),W2)) ).

% zdiv_numeral_Bit1
tff(fact_2746_semiring__norm_I16_J,axiom,
    ! [M2: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M2)),aa(num,num,bit1,N)) = 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),M2),N)),bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M2),N)))) ).

% semiring_norm(16)
tff(fact_2747_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_2748_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_2749_sin__npi2,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),N))) = zero_zero(real) ).

% sin_npi2
tff(fact_2750_sin__npi,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = zero_zero(real) ).

% sin_npi
tff(fact_2751_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_2752_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_2753_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_2754_Suc__0__and__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),N) = modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Suc_0_and_eq
tff(fact_2755_and__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),N),aa(nat,nat,suc,zero_zero(nat))) = modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% and_Suc_0_eq
tff(fact_2756_div__Suc__eq__div__add3,axiom,
    ! [M2: nat,N: nat] : divide_divide(nat,M2,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = divide_divide(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% div_Suc_eq_div_add3
tff(fact_2757_Suc__div__eq__add3__div__numeral,axiom,
    ! [M2: nat,V3: num] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M2))),aa(num,nat,numeral_numeral(nat),V3)) = 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))),M2),aa(num,nat,numeral_numeral(nat),V3)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_2758_mod__Suc__eq__mod__add3,axiom,
    ! [M2: nat,N: nat] : modulo_modulo(nat,M2,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = modulo_modulo(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_2759_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M2: nat,V3: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M2))),aa(num,nat,numeral_numeral(nat),V3)) = 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))),M2),aa(num,nat,numeral_numeral(nat),V3)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_2760_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_2761_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_2762_sin__2pi__minus,axiom,
    ! [X: real] : sin(real,aa(real,real,aa(real,fun(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_2763_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_2764_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_2765_zmod__numeral__Bit1,axiom,
    ! [V3: num,W2: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,V3)),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),V3),aa(num,int,numeral_numeral(int),W2)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_2766_sin__2npi,axiom,
    ! [N: 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),N))),pi)) = zero_zero(real) ).

% sin_2npi
tff(fact_2767_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K2: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K2))) = 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,N),aa(num,int,numeral_numeral(int),K2))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_2768_sin__3over2__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(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_2769_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X23: num] : Y != bit0(X23)
       => ~ ! [X32: num] : Y != aa(num,num,bit1,X32) ) ) ).

% num.exhaust
tff(fact_2770_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = 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),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_2771_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bit0(N))) ).

% eval_nat_numeral(3)
tff(fact_2772_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,Q2: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M2)),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M2),aa(num,A,numeral_numeral(A),Q2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_2773_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,Q2: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M2)),aa(num,A,numeral_numeral(A),bit0(Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(N)),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(12)
tff(fact_2774_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,Q2: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(M2)),aa(num,A,numeral_numeral(A),bit0(Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(10)
tff(fact_2775_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = 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),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_code(3)
tff(fact_2776_power__numeral__odd,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z2: A,W2: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2))) ) ).

% power_numeral_odd
tff(fact_2777_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),bit0(Q2))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_2778_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),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),A3),A3)),A3) ) ).

% power3_eq_cube
tff(fact_2779_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_2780_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N) ).

% Suc3_eq_add_3
tff(fact_2781_num_Osize_I6_J,axiom,
    ! [X33: num] : aa(num,nat,size_size(num),aa(num,num,bit1,X33)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X33)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(6)
tff(fact_2782_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2783_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M2)),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M2),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2784_Suc__div__eq__add3__div,axiom,
    ! [M2: nat,N: nat] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M2))),N) = 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))),M2),N) ).

% Suc_div_eq_add3_div
tff(fact_2785_card__3__iff,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S3) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
    <=> ? [X4: A,Y5: A,Z3: A] :
          ( ( S3 = aa(set(A),set(A),insert(A,X4),aa(set(A),set(A),insert(A,Y5),aa(set(A),set(A),insert(A,Z3),bot_bot(set(A))))) )
          & ( X4 != Y5 )
          & ( Y5 != Z3 )
          & ( X4 != Z3 ) ) ) ).

% card_3_iff
tff(fact_2786_Suc__mod__eq__add3__mod,axiom,
    ! [M2: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M2))),N) = 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))),M2),N) ).

% Suc_mod_eq_add3_mod
tff(fact_2787_mod__exhaust__less__4,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = zero_zero(nat) )
      | ( modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = one_one(nat) )
      | ( modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
      | ( modulo_modulo(nat,M2,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_2788_m2pi__less__pi,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_2789_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),sin(real,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_2790_sin__coeff__Suc,axiom,
    ! [N: nat] : sin_coeff(aa(nat,nat,suc,N)) = divide_divide(real,cos_coeff(N),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))) ).

% sin_coeff_Suc
tff(fact_2791_sin__lt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),pi),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sin(real,X)),zero_zero(real))) ) ) ).

% sin_lt_zero
tff(fact_2792_and__nat__unfold,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ( M2 = zero_zero(nat) )
          | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M2),N) = zero_zero(nat) ) )
      & ( ~ ( ( M2 = zero_zero(nat) )
            | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),divide_divide(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ).

% and_nat_unfold
tff(fact_2793_and__nat__rec,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fconj(aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2)),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),divide_divide(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% and_nat_rec
tff(fact_2794_sin__le__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),pi),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sin(real,X)),zero_zero(real))) ) ) ).

% sin_le_zero
tff(fact_2795_cos__coeff__Suc,axiom,
    ! [N: nat] : cos_coeff(aa(nat,nat,suc,N)) = divide_divide(real,aa(real,real,uminus_uminus(real),sin_coeff(N)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))) ).

% cos_coeff_Suc
tff(fact_2796_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
     => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% odd_mod_4_div_2
tff(fact_2797_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),sin(real,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),N))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_2798_sin__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( ( sin(real,X) = zero_zero(real) )
       => ? [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N3)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_2799_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ( ? [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5))
            & ( 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),N5)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_2800_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => ? [T6: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),T6))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T6),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_gc(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_2801_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ? [T6: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T6))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T6),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_gc(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_2802_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),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_ga(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_2803_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: 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,K2)))) = 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,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2))),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_2804_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M2),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M2))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M2),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit1,M2),bit0(aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_2805_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M2),N))
           => ( unique8689654367752047608divmod(A,bit0(M2),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(M2))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M2),N))
           => ( unique8689654367752047608divmod(A,bit0(M2),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,bit0(M2),bit0(aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_2806_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_idempotent
tff(fact_2807_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_abs
tff(fact_2808_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_2809_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_2810_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_2811_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_2812_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : aa(A,A,abs_abs(A),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),N) ) ).

% abs_numeral
tff(fact_2813_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).

% abs_mult_self_eq
tff(fact_2814_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: 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),A3)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_add_abs
tff(fact_2815_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_2816_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_minus_cancel
tff(fact_2817_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_minus
tff(fact_2818_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: A,K2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M2),aa(A,A,abs_abs(A),K2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M2),K2)) ) ) ).

% dvd_abs_iff
tff(fact_2819_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M2: A,K2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,abs_abs(A),M2)),K2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M2),K2)) ) ) ).

% abs_dvd_iff
tff(fact_2820_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% abs_of_nat
tff(fact_2821_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: bool] : aa(A,A,abs_abs(A),aa(bool,A,zero_neq_one_of_bool(A),P2)) = aa(bool,A,zero_neq_one_of_bool(A),P2) ) ).

% abs_bool_eq
tff(fact_2822_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_nonneg
tff(fact_2823_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),A3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% abs_le_self_iff
tff(fact_2824_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),zero_zero(A)))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_2825_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A3)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_2826_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),N) ) ).

% abs_neg_numeral
tff(fact_2827_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_2828_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_2829_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2830_eq__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K2) = aa(nat,nat,suc,N) )
    <=> ( pred_numeral(K2) = N ) ) ).

% eq_numeral_Suc
tff(fact_2831_Suc__eq__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( aa(nat,nat,suc,N) = aa(num,nat,numeral_numeral(nat),K2) )
    <=> ( N = pred_numeral(K2) ) ) ).

% Suc_eq_numeral
tff(fact_2832_sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F3: fun(A,B),A6: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A6))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_gd(fun(A,B),fun(A,B),F3)),A6))) ) ).

% sum_abs
tff(fact_2833_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,A3,aa(A,A,abs_abs(A),B2))),zero_zero(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2834_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,A3,aa(A,A,abs_abs(A),B2))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2835_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
         => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_nonpos
tff(fact_2836_less__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),pred_numeral(K2))) ) ).

% less_Suc_numeral
tff(fact_2837_less__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),pred_numeral(K2)),N)) ) ).

% less_numeral_Suc
tff(fact_2838_pred__numeral__simps_I3_J,axiom,
    ! [K2: num] : pred_numeral(aa(num,num,bit1,K2)) = aa(num,nat,numeral_numeral(nat),bit0(K2)) ).

% pred_numeral_simps(3)
tff(fact_2839_le__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),pred_numeral(K2)),N)) ) ).

% le_numeral_Suc
tff(fact_2840_le__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K2)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),pred_numeral(K2))) ) ).

% le_Suc_numeral
tff(fact_2841_diff__numeral__Suc,axiom,
    ! [K2: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),pred_numeral(K2)),N) ).

% diff_numeral_Suc
tff(fact_2842_diff__Suc__numeral,axiom,
    ! [N: nat,K2: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),pred_numeral(K2)) ).

% diff_Suc_numeral
tff(fact_2843_max__Suc__numeral,axiom,
    ! [N: nat,K2: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K2)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),pred_numeral(K2))) ).

% max_Suc_numeral
tff(fact_2844_max__numeral__Suc,axiom,
    ! [K2: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K2)),N)) ).

% max_numeral_Suc
tff(fact_2845_sum__abs__ge__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F3: fun(A,B),A6: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_gd(fun(A,B),fun(A,B),F3)),A6))) ) ).

% sum_abs_ge_zero
tff(fact_2846_minus__numeral__div__numeral,axiom,
    ! [M2: num,N: num] : divide_divide(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M2,N))) ).

% minus_numeral_div_numeral
tff(fact_2847_numeral__div__minus__numeral,axiom,
    ! [M2: num,N: num] : divide_divide(int,aa(num,int,numeral_numeral(int),M2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M2,N))) ).

% numeral_div_minus_numeral
tff(fact_2848_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A3)),N)))
        <=> ( ( A3 != zero_zero(A) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_2849_sin__cos__squared__add3,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(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,X))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,X))) = one_one(A) ) ).

% sin_cos_squared_add3
tff(fact_2850_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)))
        <=> unique5940410009612947441es_aux(A,unique8689654367752047608divmod(A,N,M2)) ) ) ).

% dvd_numeral_simp
tff(fact_2851_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num] : unique8689654367752047608divmod(A,M2,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),M2)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_2852_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,bit0(N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(3)
tff(fact_2853_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_2854_one__div__minus__numeral,axiom,
    ! [N: num] : divide_divide(int,one_one(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% one_div_minus_numeral
tff(fact_2855_minus__one__div__numeral,axiom,
    ! [N: num] : divide_divide(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% minus_one_div_numeral
tff(fact_2856_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_2857_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_2858_cos__2pi__minus,axiom,
    ! [X: real] : cos(real,aa(real,real,aa(real,fun(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_2859_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),bit0(K2))) = 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),K2))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_numeral_bit0
tff(fact_2860_cos__npi,axiom,
    ! [N: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% cos_npi
tff(fact_2861_cos__npi2,axiom,
    ! [N: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),N))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% cos_npi2
tff(fact_2862_cos__2npi,axiom,
    ! [N: 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),N))),pi)) = one_one(real) ).

% cos_2npi
tff(fact_2863_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: 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(K2)))) = 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),K2)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_2864_cos__3over2__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(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_2865_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K2: 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,K2))) = 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),K2))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_2866_cos__pi__eq__zero,axiom,
    ! [M2: nat] : cos(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))),M2)))),aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ).

% cos_pi_eq_zero
tff(fact_2867_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,abs_abs(A),A3))) ) ).

% abs_ge_self
tff(fact_2868_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% abs_le_D1
tff(fact_2869_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_2870_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_mult
tff(fact_2871_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2872_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ).

% abs_minus_commute
tff(fact_2873_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_2874_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K2: A] :
          ( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),L),K2)) ) ) ).

% dvd_if_abs_eq
tff(fact_2875_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A3))) ) ).

% abs_ge_zero
tff(fact_2876_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),zero_zero(A))) ) ).

% abs_not_less_zero
tff(fact_2877_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_pos
tff(fact_2878_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))) ) ).

% abs_triangle_ineq
tff(fact_2879_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),B2)),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D3))) ) ) ) ).

% abs_mult_less
tff(fact_2880_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)))) ) ).

% abs_triangle_ineq2_sym
tff(fact_2881_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))) ) ).

% abs_triangle_ineq3
tff(fact_2882_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))) ) ).

% abs_triangle_ineq2
tff(fact_2883_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),divide_divide(A,A3,B2)) = divide_divide(A,aa(A,A,abs_abs(A),A3),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_2884_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)) ) ) ) ).

% abs_leI
tff(fact_2885_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ).

% abs_le_D2
tff(fact_2886_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ) ).

% abs_le_iff
tff(fact_2887_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,abs_abs(A),A3))) ) ).

% abs_ge_minus_self
tff(fact_2888_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ) ).

% abs_less_iff
tff(fact_2889_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R3: real,A5: real] :
      ( ( X = aa(real,real,aa(real,fun(real,real),times_times(real),R3),cos(real,A5)) )
      & ( Y = aa(real,real,aa(real,fun(real,real),times_times(real),R3),sin(real,A5)) ) ) ).

% polar_Ex
tff(fact_2890_numeral__eq__Suc,axiom,
    ! [K2: num] : aa(num,nat,numeral_numeral(nat),K2) = aa(nat,nat,suc,pred_numeral(K2)) ).

% numeral_eq_Suc
tff(fact_2891_sin__cos__le1,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),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_2892_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [X: A] :
          ( ! [E2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),E2)) )
         => ( X = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2893_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).

% abs_mult_pos
tff(fact_2894_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A3: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
              | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
              | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% abs_eq_mult
tff(fact_2895_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_2896_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A3))),zero_zero(A))) ) ).

% abs_minus_le_zero
tff(fact_2897_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,abs_abs(A),B2) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
            & ( ( B2 = A3 )
              | ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2898_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A3) = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
            & ( ( A3 = B2 )
              | ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2899_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( divide_divide(A,aa(A,A,abs_abs(A),X),Y) = aa(A,A,abs_abs(A),divide_divide(A,X,Y)) ) ) ) ).

% abs_div_pos
tff(fact_2900_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A3)),N))) ) ).

% zero_le_power_abs
tff(fact_2901_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X5: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X5) = aa(A,A,uminus_uminus(A),X5) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X5) = X5 ) ) ) ) ).

% abs_if_raw
tff(fact_2902_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ) ).

% abs_if
tff(fact_2903_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_neg
tff(fact_2904_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sin(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),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),sin(A,Y))) ) ).

% sin_add
tff(fact_2905_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A3: A,R: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3))),R))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R))) ) ) ) ).

% abs_diff_le_iff
tff(fact_2906_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A,C3: A,D3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),D3)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C3))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3))))) ) ).

% abs_diff_triangle_ineq
tff(fact_2907_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))) ) ).

% abs_triangle_ineq4
tff(fact_2908_sin__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),sin(A,Y))) ) ).

% sin_diff
tff(fact_2909_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A3: A,R: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3))),R))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R))) ) ) ) ).

% abs_diff_less_iff
tff(fact_2910_pred__numeral__def,axiom,
    ! [K2: num] : pred_numeral(K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K2)),one_one(nat)) ).

% pred_numeral_def
tff(fact_2911_lessThan__nat__numeral,axiom,
    ! [K2: num] : aa(nat,set(nat),set_ord_lessThan(nat),aa(num,nat,numeral_numeral(nat),K2)) = aa(set(nat),set(nat),insert(nat,pred_numeral(K2)),aa(nat,set(nat),set_ord_lessThan(nat),pred_numeral(K2))) ).

% lessThan_nat_numeral
tff(fact_2912_atMost__nat__numeral,axiom,
    ! [K2: num] : aa(nat,set(nat),set_ord_atMost(nat),aa(num,nat,numeral_numeral(nat),K2)) = aa(set(nat),set(nat),insert(nat,aa(num,nat,numeral_numeral(nat),K2)),aa(nat,set(nat),set_ord_atMost(nat),pred_numeral(K2))) ).

% atMost_nat_numeral
tff(fact_2913_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y))) ) ).

% cos_add
tff(fact_2914_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y))) ) ).

% cos_diff
tff(fact_2915_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),X)))) ) ).

% abs_add_one_gt_zero
tff(fact_2916_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_2917_fact__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K2: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K2)),semiring_char_0_fact(A,pred_numeral(K2))) ) ).

% fact_numeral
tff(fact_2918_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_2919_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),aa(A,A,abs_abs(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_le_square_iff
tff(fact_2920_divmod__int__def,axiom,
    ! [M2: num,N: num] : unique8689654367752047608divmod(int,M2,N) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),divide_divide(int,aa(num,int,numeral_numeral(int),M2),aa(num,int,numeral_numeral(int),N))),modulo_modulo(int,aa(num,int,numeral_numeral(int),M2),aa(num,int,numeral_numeral(int),N))) ).

% divmod_int_def
tff(fact_2921_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: fun(A,fun(A,bool)),X: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3))
             => pp(aa(A,bool,aa(A,fun(A,bool),P2,X3),aa(nat,A,aa(A,fun(nat,A),power_power(A),X3),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),P2,aa(A,A,abs_abs(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_sqrt_wlog
tff(fact_2922_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),Y)) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2923_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_le_1
tff(fact_2924_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] : unique8689654367752047608divmod(A,M2,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),divide_divide(A,aa(num,A,numeral_numeral(A),M2),aa(num,A,numeral_numeral(A),N))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M2),aa(num,A,numeral_numeral(A),N))) ) ).

% divmod_def
tff(fact_2925_divmod_H__nat__def,axiom,
    ! [M2: num,N: num] : unique8689654367752047608divmod(nat,M2,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),divide_divide(nat,aa(num,nat,numeral_numeral(nat),M2),aa(num,nat,numeral_numeral(nat),N))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),M2),aa(num,nat,numeral_numeral(nat),N))) ).

% divmod'_nat_def
tff(fact_2926_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono_even
tff(fact_2927_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),X: fun(A,B),A3: fun(A,B),B2: B,Delta: B] :
          ( ! [I3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I3))) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),X),I5) = one_one(B) )
           => ( ! [I3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),I5))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,I3)),B2))),Delta)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_ge(fun(A,B),fun(fun(A,B),fun(A,B)),X),A3)),I5)),B2))),Delta)) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_2928_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,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,divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),Z2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_plus_cos
tff(fact_2929_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)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),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_2930_cos__double__less__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(num,real,numeral_numeral(real),bit0(one2))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),X))),one_one(real))) ) ) ).

% cos_double_less_one
tff(fact_2931_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,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,W2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(A)) ) ).

% cos_double_cos
tff(fact_2932_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,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(bit0(one2)))),aa(nat,A,aa(A,fun(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_2933_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z2: A] : aa(A,A,aa(A,fun(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,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,divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z2),W2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_diff_cos
tff(fact_2934_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z2: A] : aa(A,A,aa(A,fun(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,divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),Z2),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_diff_sin
tff(fact_2935_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,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,divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),Z2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_plus_sin
tff(fact_2936_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)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z2))),sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),Z2))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_sin
tff(fact_2937_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)) = 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,aa(A,fun(A,A),minus_minus(A),W2),Z2))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_cos
tff(fact_2938_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)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),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_2939_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,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_double
tff(fact_2940_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),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_gc(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_2941_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ( ? [X4: 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),X4)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)
        | ? [X4: 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),X4)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_2942_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,aa(A,fun(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,aa(A,fun(nat,A),power_power(A),sin(A,W2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% cos_double_sin
tff(fact_2943_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N))
           => ( unique8689654367752047608divmod(A,M2,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M2)) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N))
           => ( unique8689654367752047608divmod(A,M2,N) = unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M2,bit0(N))) ) ) ) ) ).

% divmod_divmod_step
tff(fact_2944_sin__expansion__lemma,axiom,
    ! [X: real,M2: nat] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),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,M2))),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) = cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M2)),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% sin_expansion_lemma
tff(fact_2945_cos__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( ( cos(real,X) = zero_zero(real) )
       => ? [N3: nat] :
            ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N3)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_2946_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( cos(real,X) = zero_zero(real) )
    <=> ( ? [N5: nat] :
            ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N5: nat] :
            ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5))
            & ( 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),N5)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_2947_cos__expansion__lemma,axiom,
    ! [X: real,M2: nat] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),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,M2))),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),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M2)),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% cos_expansion_lemma
tff(fact_2948_cos__paired,axiom,
    ! [X: real] : sums(real,aTP_Lamp_gf(real,fun(nat,real),X),cos(real,X)) ).

% cos_paired
tff(fact_2949_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ? [T6: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T6))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))
          & ( X = cos(real,T6) )
          & ( Y = sin(real,T6) ) ) ) ).

% sincos_total_2pi_le
tff(fact_2950_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ~ ! [T6: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T6))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))
             => ( ( X = cos(real,T6) )
               => ( Y != sin(real,T6) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_2951_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => topological_monoseq(real,aTP_Lamp_gg(real,fun(nat,real),X)) ) ).

% monoseq_arctan_series
tff(fact_2952_summable__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => summable(real,aTP_Lamp_gh(real,fun(nat,real),X)) ) ).

% summable_arctan_series
tff(fact_2953_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(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,aa(real,fun(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_2954_pi__series,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = suminf(real,aTP_Lamp_gi(nat,real)) ).

% pi_series
tff(fact_2955_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)) = 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,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_2956_and__int_Opsimps,axiom,
    ! [K2: int,L: int] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K2),L))
     => ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K2)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))) ) )
        & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
              & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K2)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_2957_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_2958_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I2: nat,F3: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cj(nat,fun(fun(nat,A),fun(nat,A)),I2),F3)) ) ).

% summable_single
tff(fact_2959_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_bz(nat,A)) ) ).

% summable_zero
tff(fact_2960_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_gj(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_2961_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( ln_ln(A,one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_2962_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cp(A,fun(fun(nat,A),fun(nat,A)),C3),F3))
        <=> ( ( C3 = zero_zero(A) )
            | summable(A,F3) ) ) ) ).

% summable_cmult_iff
tff(fact_2963_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gk(fun(nat,A),fun(A,fun(nat,A)),F3),C3))
        <=> ( ( C3 = zero_zero(A) )
            | summable(A,F3) ) ) ) ).

% summable_divide_iff
tff(fact_2964_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A6: set(nat),F3: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cb(set(nat),fun(fun(nat,A),fun(nat,A)),A6),F3)) ) ) ).

% summable_If_finite_set
tff(fact_2965_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P2: fun(nat,bool),F3: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),P2)))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cc(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),P2),F3)) ) ) ).

% summable_If_finite
tff(fact_2966_tan__npi,axiom,
    ! [N: 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),N)),pi)) = zero_zero(real) ).

% tan_npi
tff(fact_2967_tan__periodic__n,axiom,
    ! [X: real,N: 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),N)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_n
tff(fact_2968_tan__periodic__nat,axiom,
    ! [X: real,N: 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),N)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_nat
tff(fact_2969_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F3: fun(nat,A)] : suminf(A,aTP_Lamp_gl(fun(nat,A),fun(nat,A),F3)) = aa(nat,A,F3,zero_zero(nat)) ) ).

% powser_zero
tff(fact_2970_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_2971_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F3,N3)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),suminf(A,F3))) ) ) ) ).

% suminf_pos
tff(fact_2972_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N3)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),suminf(A,F3))) ) ) ) ).

% suminf_nonneg
tff(fact_2973_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N3)))
           => ( ( suminf(A,F3) = zero_zero(A) )
            <=> ! [N5: nat] : aa(nat,A,F3,N5) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_2974_suminf__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,F3)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,A)),F3),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),suminf(A,F3)) ) ) ) ).

% suminf_mult
tff(fact_2975_suminf__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,F3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,F3)),C3) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),F3),C3)) ) ) ) ).

% suminf_mult2
tff(fact_2976_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,A)] :
          ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N3)),aa(nat,A,G3,N3)))
         => ( summable(A,F3)
           => ( summable(A,G3)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),suminf(A,F3)),suminf(A,G3))) ) ) ) ) ).

% suminf_le
tff(fact_2977_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N3)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),suminf(A,F3)))
            <=> ? [I: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F3,I))) ) ) ) ) ).

% suminf_pos_iff
tff(fact_2978_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),I2: nat] :
          ( summable(A,F3)
         => ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N3)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F3,I2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),suminf(A,F3))) ) ) ) ) ).

% suminf_pos2
tff(fact_2979_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( ? [N8: nat] :
            ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N3))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N3))),aa(nat,real,G3,N3))) )
         => ( summable(real,G3)
           => summable(A,F3) ) ) ) ).

% summable_comparison_test
tff(fact_2980_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G3: fun(nat,real),N6: nat,F3: fun(nat,A)] :
          ( summable(real,G3)
         => ( ! [N3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N3))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N3))),aa(nat,real,G3,N3))) )
           => summable(A,F3) ) ) ) ).

% summable_comparison_test'
tff(fact_2981_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C3: A] :
          ( summable(A,aTP_Lamp_gm(A,fun(nat,A),C3))
        <=> ( C3 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_2982_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),X: A] :
          ( summable(A,F3)
         => ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),N3))),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),suminf(A,F3)),X)) ) ) ) ).

% suminf_le_const
tff(fact_2983_summable__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,F3)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),F3),C3)) ) ) ).

% summable_mult2
tff(fact_2984_summable__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,F3)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,A)),F3),C3)) ) ) ).

% summable_mult
tff(fact_2985_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,aTP_Lamp_cq(fun(nat,A),fun(nat,A),F3))
        <=> summable(A,F3) ) ) ).

% summable_Suc_iff
tff(fact_2986_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( suminf(A,aTP_Lamp_cq(fun(nat,A),fun(nat,A),F3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F3)),aa(nat,A,F3,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_2987_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),I5: set(nat)] :
          ( summable(A,F3)
         => ( pp(aa(set(nat),bool,finite_finite2(nat),I5))
           => ( ! [N3: nat] :
                  ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N3),aa(set(nat),set(nat),uminus_uminus(set(nat)),I5)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),I5)),suminf(A,F3))) ) ) ) ) ).

% sum_le_suminf
tff(fact_2988_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),X: A,Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fk(fun(nat,A),fun(A,fun(nat,A)),F3),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),real_V7770717601297561774m_norm(A,X)))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_gn(fun(nat,A),fun(A,fun(nat,real)),F3),Z2)) ) ) ) ).

% powser_insidea
tff(fact_2989_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N6: set(nat),F3: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),N6))
         => ( ! [N3: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N3),N6))
               => ( aa(nat,A,F3,N3) = zero_zero(A) ) )
           => summable(A,F3) ) ) ) ).

% summable_finite
tff(fact_2990_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cp(A,fun(fun(nat,A),fun(nat,A)),C3),F3))
         => ( ( C3 != zero_zero(A) )
           => summable(A,F3) ) ) ) ).

% summable_mult_D
tff(fact_2991_abs__zmult__eq__1,axiom,
    ! [M2: int,N: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),M2),N)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),M2) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_2992_summable__zero__power,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aa(A,fun(nat,A),power_power(A),zero_zero(A))) ) ).

% summable_zero_power
tff(fact_2993_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),N: nat] :
          ( summable(A,F3)
         => ( ! [M: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F3,M))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),N))),suminf(A,F3))) ) ) ) ).

% sum_less_suminf
tff(fact_2994_abs__div,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Y),X))
     => ( aa(int,int,abs_abs(int),divide_divide(int,X,Y)) = divide_divide(int,aa(int,int,abs_abs(int),X),aa(int,int,abs_abs(int),Y)) ) ) ).

% abs_div
tff(fact_2995_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_go(fun(nat,A),fun(A,fun(nat,A)),F3),Z2))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_go(fun(nat,A),fun(A,fun(nat,A)),F3),Z2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,F3,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gp(fun(nat,A),fun(A,fun(nat,A)),F3),Z2))),Z2)) ) ) ) ).

% powser_split_head(1)
tff(fact_2996_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_go(fun(nat,A),fun(A,fun(nat,A)),F3),Z2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gp(fun(nat,A),fun(A,fun(nat,A)),F3),Z2))),Z2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_go(fun(nat,A),fun(A,fun(nat,A)),F3),Z2))),aa(nat,A,F3,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_2997_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R: real,F3: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
         => ( summable(A,F3)
           => ? [N9: nat] :
              ! [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),N4))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_gq(fun(nat,A),fun(nat,fun(nat,A)),F3),N4)))),R)) ) ) ) ) ).

% suminf_exist_split
tff(fact_2998_summable__Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_gr(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_gr(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gt(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_2999_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_gr(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_gr(fun(nat,A),fun(nat,real),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A3)),suminf(A,B2)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gt(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_3000_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),N: nat,I2: nat] :
          ( summable(A,F3)
         => ( ! [M: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,M))) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),I2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F3,I2)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),N))),suminf(A,F3))) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_3001_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_gr(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_gr(fun(nat,A),fun(nat,real),B2))
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gt(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B2),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A3)),suminf(A,B2))) ) ) ) ).

% Cauchy_product_sums
tff(fact_3002_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A)] : summable(A,aTP_Lamp_gu(fun(nat,A),fun(nat,A),F3)) ) ).

% summable_zero_power'
tff(fact_3003_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A)] : summable(A,aTP_Lamp_ay(fun(nat,A),fun(nat,A),F3)) ) ).

% summable_0_powser
tff(fact_3004_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_go(fun(nat,A),fun(A,fun(nat,A)),F3),Z2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_gp(fun(nat,A),fun(A,fun(nat,A)),F3),Z2)) ) ) ).

% powser_split_head(3)
tff(fact_3005_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gv(fun(nat,A),fun(A,fun(nat,A)),F3),Z2))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_fk(fun(nat,A),fun(A,fun(nat,A)),F3),Z2)) ) ) ).

% summable_powser_split_head
tff(fact_3006_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),M2: nat,Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_gw(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F3),M2),Z2))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_fk(fun(nat,A),fun(A,fun(nat,A)),F3),Z2)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_3007_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( ? [N8: nat] :
            ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N3))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N3))),aa(nat,real,G3,N3))) )
         => ( summable(real,G3)
           => summable(real,aTP_Lamp_gx(fun(nat,A),fun(nat,real),F3)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_3008_summable__rabs__comparison__test,axiom,
    ! [F3: fun(nat,real),G3: fun(nat,real)] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N3))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,F3,N3))),aa(nat,real,G3,N3))) )
     => ( summable(real,G3)
       => summable(real,aTP_Lamp_gy(fun(nat,real),fun(nat,real),F3)) ) ) ).

% summable_rabs_comparison_test
tff(fact_3009_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N6: set(nat),F3: fun(nat,A)] :
          ( pp(aa(set(nat),bool,finite_finite2(nat),N6))
         => ( ! [N3: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N3),N6))
               => ( aa(nat,A,F3,N3) = zero_zero(A) ) )
           => ( suminf(A,F3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),N6) ) ) ) ) ).

% suminf_finite
tff(fact_3010_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => ( 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),ln_ln(real,X)),ln_ln(real,Y)) ) ) ) ).

% ln_mult
tff(fact_3011_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),X: A,Z2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_go(fun(nat,A),fun(A,fun(nat,A)),F3),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),real_V7770717601297561774m_norm(A,X)))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_go(fun(nat,A),fun(A,fun(nat,A)),F3),Z2)) ) ) ) ).

% powser_inside
tff(fact_3012_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),X: A] :
          ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N3)))
         => ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),N3))),X))
           => summable(A,F3) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_3013_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A3: fun(nat,A),B6: A] :
          ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,A3,N3)))
         => ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A3),aa(nat,set(nat),set_ord_atMost(nat),N3))),B6))
           => summable(A,A3) ) ) ) ).

% bounded_imp_summable
tff(fact_3014_zdvd__mult__cancel1,axiom,
    ! [M2: int,N: int] :
      ( ( M2 != zero_zero(int) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),M2),N)),M2))
      <=> ( aa(int,int,abs_abs(int),N) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_3015_sum__pos__lt__pair,axiom,
    ! [F3: fun(nat,real),K2: nat] :
      ( summable(real,F3)
     => ( ! [D2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D2)))),aa(nat,real,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D2)),one_one(nat)))))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),F3),aa(nat,set(nat),set_ord_lessThan(nat),K2))),suminf(real,F3))) ) ) ).

% sum_pos_lt_pair
tff(fact_3016_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ln_ln(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),ln_ln(real,X)) ) ) ).

% ln_realpow
tff(fact_3017_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),E3: real] :
          ( summable(A,F3)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
           => ~ ! [N9: nat] :
                  ~ ! [M3: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),M3))
                     => ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,M3,N4)))),E3)) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_3018_summable__power__series,axiom,
    ! [F3: fun(nat,real),Z2: real] :
      ( ! [I3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,I3)),one_one(real)))
     => ( ! [I3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,F3,I3)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Z2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),one_one(real)))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_gz(fun(nat,real),fun(real,fun(nat,real)),F3),Z2)) ) ) ) ) ).

% summable_power_series
tff(fact_3019_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R: real,R0: real,A3: fun(nat,A),M5: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),R),R0))
           => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A3,N3))),aa(nat,real,aa(real,fun(nat,real),power_power(real),R0),N3))),M5))
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_ha(real,fun(fun(nat,A),fun(nat,real)),R),A3)) ) ) ) ) ).

% Abel_lemma
tff(fact_3020_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F3: fun(nat,int),K2: int] :
      ( ! [I3: nat] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),I3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N)) )
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,suc,I3))),aa(nat,int,F3,I3)))),one_one(int))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F3,M2)),K2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),aa(nat,int,F3,N)))
           => ? [I3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),I3))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
                & ( aa(nat,int,F3,I3) = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_3021_incr__lemma,axiom,
    ! [D3: int,Z2: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => pp(aa(int,bool,aa(int,fun(int,bool),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,aa(int,fun(int,int),minus_minus(int),X),Z2))),one_one(int))),D3)))) ) ).

% incr_lemma
tff(fact_3022_decr__lemma,axiom,
    ! [D3: int,X: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Z2))),one_one(int))),D3))),Z2)) ) ).

% decr_lemma
tff(fact_3023_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C3: real,N6: nat,F3: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C3),one_one(real)))
         => ( ! [N3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N3))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,aa(nat,nat,suc,N3)))),aa(real,real,aa(real,fun(real,real),times_times(real),C3),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N3))))) )
           => summable(A,F3) ) ) ) ).

% summable_ratio_test
tff(fact_3024_ln__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( ln_ln(real,X) = suminf(real,aTP_Lamp_hb(real,fun(nat,real),X)) ) ) ) ).

% ln_series
tff(fact_3025_nat__ivt__aux,axiom,
    ! [N: nat,F3: fun(nat,int),K2: int] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,suc,I3))),aa(nat,int,F3,I3)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K2))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),aa(nat,int,F3,N)))
         => ? [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
              & ( aa(nat,int,F3,I3) = K2 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_3026_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)) = divide_divide(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_3027_nat0__intermed__int__val,axiom,
    ! [N: nat,F3: fun(nat,int),K2: int] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat)))),aa(nat,int,F3,I3)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K2))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K2),aa(nat,int,F3,N)))
         => ? [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
              & ( aa(nat,int,F3,I3) = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_3028_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_3029_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3030_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y))) = divide_divide(A,cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3031_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tan(A),X) = 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_3032_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,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,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),ln_ln(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),X)))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_3033_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),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,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_3034_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa2) = Y )
     => ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2))
       => ~ ( ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                  & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa2))))) ) )
              & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,X,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,Xa2,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) )
           => ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_3035_arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( arctan(X) = suminf(real,aTP_Lamp_gh(real,fun(nat,real),X)) ) ) ).

% arctan_series
tff(fact_3036_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C3: fun(nat,A),X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hc(fun(nat,A),fun(A,fun(nat,A)),C3),X))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_hd(fun(nat,A),fun(A,fun(nat,A)),C3),X),suminf(A,aa(A,fun(nat,A),aTP_Lamp_hc(fun(nat,A),fun(A,fun(nat,A)),C3),X))) ) ) ).

% diffs_equiv
tff(fact_3037_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero(real) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ? [T6: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T6)))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),X)))
            & ( exp(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_he(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,exp(real,T6),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_3038_arctan__double,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),arctan(X)) = arctan(divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),X),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% arctan_double
tff(fact_3039_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( topological_monoseq(A,X6)
        <=> ( ! [M6: nat,N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M6),N5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,M6)),aa(nat,A,X6,N5))) )
            | ! [M6: nat,N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M6),N5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N5)),aa(nat,A,X6,M6))) ) ) ) ) ).

% monoseq_def
tff(fact_3040_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( exp(A,zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_3041_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : exp(A,X) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_3042_exp__times__arg__commute,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [A6: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,A6)),A6) = aa(A,A,aa(A,fun(A,A),times_times(A),A6),exp(A,A6)) ) ).

% exp_times_arg_commute
tff(fact_3043_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,X)),exp(A,Y)) = exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) ) ).

% mult_exp_exp
tff(fact_3044_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y: 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),Y),X) )
         => ( exp(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),exp(A,X)),exp(A,Y)) ) ) ) ).

% exp_add_commuting
tff(fact_3045_exp__minus__inverse,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,X)),exp(A,aa(A,A,uminus_uminus(A),X))) = one_one(A) ) ).

% exp_minus_inverse
tff(fact_3046_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [N: nat,X: A] : exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),X)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,X)),N) ) ).

% exp_of_nat_mult
tff(fact_3047_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,N: nat] : exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(nat,A,semiring_1_of_nat(A),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,X)),N) ) ).

% exp_of_nat2_mult
tff(fact_3048_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C3: fun(nat,A),X5: nat] : aa(nat,A,diffs(A,C3),X5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X5))),aa(nat,A,C3,aa(nat,nat,suc,X5))) ) ).

% diffs_def
tff(fact_3049_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),X: A] :
          ( ! [X3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),C3),X3))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_hg(fun(nat,A),fun(A,fun(nat,A)),C3),X)) ) ) ).

% termdiff_converges_all
tff(fact_3050_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,divide_divide(A,X,aa(nat,A,semiring_1_of_nat(A),N)))),N) = exp(A,X) ) ) ) ).

% exp_divide_power_eq
tff(fact_3051_exp__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z2: 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,aa(A,fun(nat,A),power_power(A),exp(A,Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% exp_double
tff(fact_3052_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,K5: real,C3: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),K5))
         => ( ! [X3: A] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X3)),K5))
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),C3),X3)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_hh(A,fun(fun(nat,A),fun(nat,A)),X),C3)) ) ) ) ).

% termdiff_converges
tff(fact_3053_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(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_3054_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),N))),X))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),divide_divide(real,X,aa(nat,real,semiring_1_of_nat(real),N)))),N)),exp(real,X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_3055_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),divide_divide(real,X,aa(nat,real,semiring_1_of_nat(real),N)))),N)),exp(real,aa(real,real,uminus_uminus(real),X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_3056_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),arctan(X)),arctan(Y)) = arctan(divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)))) ) ) ) ).

% arctan_add
tff(fact_3057_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z2)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(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_3058_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),X)))
      & ( exp(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_he(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,exp(real,T6),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_3059_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)))),arctan(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))),arctan(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))))))))))) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% machin_Euler
tff(fact_3060_machin,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = aa(real,real,aa(real,fun(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)))),arctan(divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2))))))),arctan(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_3061_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N3)),aa(nat,A,X6,aa(nat,nat,suc,N3))))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI1
tff(fact_3062_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N3))),aa(nat,A,X6,N3)))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI2
tff(fact_3063_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( topological_monoseq(A,X6)
        <=> ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N5)),aa(nat,A,X6,aa(nat,nat,suc,N5))))
            | ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N5))),aa(nat,A,X6,N5))) ) ) ) ).

% monoseq_Suc
tff(fact_3064_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [M: nat,N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,M)),aa(nat,A,X6,N3))) )
         => topological_monoseq(A,X6) ) ) ).

% monoI1
tff(fact_3065_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [M: nat,N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N3)),aa(nat,A,X6,M))) )
         => topological_monoseq(A,X6) ) ) ).

% monoI2
tff(fact_3066_tanh__real__altdef,axiom,
    ! [X: real] : aa(real,real,tanh(real),X) = divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(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)),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_3067_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_hj(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_3068_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),sin(real,X)),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ga(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),X)),N)))) ).

% Maclaurin_sin_bound
tff(fact_3069_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_hl(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_3070_divmod__BitM__2__eq,axiom,
    ! [M2: num] : unique8689654367752047608divmod(int,bitM(M2),bit0(one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M2)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_3071_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,X: A,B2: real] :
          ( ( aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X) = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),B2),X) )
        <=> ( ( A3 = B2 )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_3072_scaleR__zero__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real] : aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% scaleR_zero_right
tff(fact_3073_inverse__nonzero__iff__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_3074_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_3075_mult__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [A3: real,X: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X)),Y) = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) ) ).

% mult_scaleR_left
tff(fact_3076_mult__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [X: A,A3: real,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),Y)) = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) ) ).

% mult_scaleR_right
tff(fact_3077_inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)) ) ).

% inverse_mult_distrib
tff(fact_3078_scaleR__scaleR,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,B2: real,X: A] : aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),B2),X)) = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,aa(real,fun(real,real),times_times(real),A3),B2)),X) ) ).

% scaleR_scaleR
tff(fact_3079_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_3080_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_3081_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_3082_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% inverse_positive_iff_positive
tff(fact_3083_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% inverse_negative_iff_negative
tff(fact_3084_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_3085_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_3086_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),zero_zero(real)),X) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_3087_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,X: A] :
          ( ( aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(real) )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_3088_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_3089_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_3090_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_3091_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z2)) ) ) ).

% of_int_le_iff
tff(fact_3092_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_3093_dbl__dec__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K2)) = aa(num,A,numeral_numeral(A),bitM(K2)) ) ).

% dbl_dec_simps(5)
tff(fact_3094_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_3095_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_3096_left__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).

% left_inverse
tff(fact_3097_right__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),A3)) = one_one(A) ) ) ) ).

% right_inverse
tff(fact_3098_inverse__eq__divide__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num] : aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),W2)) = divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),W2)) ) ).

% inverse_eq_divide_numeral
tff(fact_3099_norm__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: real,X: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,abs_abs(real),A3)),real_V7770717601297561774m_norm(A,X)) ) ).

% norm_scaleR
tff(fact_3100_pred__numeral__simps_I2_J,axiom,
    ! [K2: num] : pred_numeral(bit0(K2)) = aa(num,nat,numeral_numeral(nat),bitM(K2)) ).

% pred_numeral_simps(2)
tff(fact_3101_inverse__eq__divide__neg__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = divide_divide(A,one_one(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) ).

% inverse_eq_divide_neg_numeral
tff(fact_3102_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),zero_zero(int))) ) ) ).

% of_int_le_0_iff
tff(fact_3103_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2)) ) ) ).

% of_int_0_le_iff
tff(fact_3104_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),zero_zero(int))) ) ) ).

% of_int_less_0_iff
tff(fact_3105_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2)) ) ) ).

% of_int_0_less_iff
tff(fact_3106_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),N)),Z2)) ) ) ).

% of_int_numeral_le_iff
tff(fact_3107_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_le_numeral_iff
tff(fact_3108_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z2)) ) ) ).

% of_int_1_le_iff
tff(fact_3109_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),one_one(int))) ) ) ).

% of_int_le_1_iff
tff(fact_3110_scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,W2: num,A3: A] : aa(A,A,aa(real,fun(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)),A3)) = aa(A,A,aa(real,fun(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))),A3) ) ).

% scaleR_times
tff(fact_3111_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W2: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2))) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_3112_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2)),X)) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_3113_sin__npi__int,axiom,
    ! [N: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),N))) = zero_zero(real) ).

% sin_npi_int
tff(fact_3114_tan__periodic__int,axiom,
    ! [X: real,I2: int] : 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(int,real,ring_1_of_int(real),I2)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_int
tff(fact_3115_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V3: num,W2: num,A3: A] : aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),V3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),A3)) = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),divide_divide(real,aa(num,real,numeral_numeral(real),W2),aa(num,real,numeral_numeral(real),V3))),A3) ) ).

% inverse_scaleR_times
tff(fact_3116_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V3: num,W2: num,A3: A] : aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),divide_divide(real,aa(num,real,numeral_numeral(real),U),aa(num,real,numeral_numeral(real),V3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),A3)) = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),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),V3))),A3) ) ).

% fraction_scaleR_times
tff(fact_3117_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_3118_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_3119_sin__int__2pin,axiom,
    ! [N: 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),N))) = zero_zero(real) ).

% sin_int_2pin
tff(fact_3120_cos__int__2pin,axiom,
    ! [N: 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),N))) = one_one(real) ).

% cos_int_2pin
tff(fact_3121_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A3)) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_3122_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_3123_cos__npi__int,axiom,
    ! [N: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
       => ( cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),N))) = one_one(real) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
       => ( cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),N))) = aa(real,real,uminus_uminus(real),one_one(real)) ) ) ) ).

% cos_npi_int
tff(fact_3124_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,X: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_3125_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z))) ) ).

% ex_le_of_int
tff(fact_3126_nonzero__norm__inverse,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),A3)) = aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,A3)) ) ) ) ).

% nonzero_norm_inverse
tff(fact_3127_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa2: 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),Xa2))),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),Xa2))) ) ).

% mult_inverse_of_int_commute
tff(fact_3128_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A3: real,X: A] :
          ( ( A3 != zero_zero(real) )
         => ( ( X != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X)) = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),A3)),aa(A,A,inverse_inverse(A),X)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_3129_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_3130_inverse__zero__imp__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
         => ( A3 = zero_zero(A) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_3131_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B2) )
         => ( ( A3 != zero_zero(A) )
           => ( ( B2 != zero_zero(A) )
             => ( A3 = B2 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_3132_nonzero__inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A3)) = A3 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_3133_nonzero__imp__inverse__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) != zero_zero(A) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_3134_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,A3: real,B2: real] :
          ( ( X != zero_zero(A) )
         => ( ( aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X) = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),B2),X) )
           => ( A3 = B2 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_3135_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_3136_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),X)) ) ).

% ex_of_int_less
tff(fact_3137_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z))) ) ).

% ex_less_of_int
tff(fact_3138_real__scaleR__def,axiom,
    ! [A3: real,X: real] : aa(real,real,aa(real,fun(real,real),real_V8093663219630862766scaleR(real),A3),X) = aa(real,real,aa(real,fun(real,real),times_times(real),A3),X) ).

% real_scaleR_def
tff(fact_3139_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C3),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),C3)),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),A3))) ) ) ) ).

% neg_le_divideR_eq
tff(fact_3140_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A3: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C3),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),C3)),B2)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),A3)),B2)) ) ) ) ).

% neg_divideR_le_eq
tff(fact_3141_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),C3)),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),A3)),B2)) ) ) ) ).

% pos_le_divideR_eq
tff(fact_3142_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A3: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),C3)),B2)),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),A3))) ) ) ) ).

% pos_divideR_le_eq
tff(fact_3143_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A3: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C3),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),C3)),B2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),A3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_3144_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C3),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),C3)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),A3))) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_3145_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A3: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),C3)),B2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),A3))) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_3146_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),C3)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),A3)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_3147_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ).

% positive_imp_inverse_positive
tff(fact_3148_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A))) ) ) ).

% negative_imp_inverse_negative
tff(fact_3149_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)))
         => ( ( A3 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_3150_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
         => ( ( A3 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_3151_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_3152_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_3153_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% less_imp_inverse_less
tff(fact_3154_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).

% inverse_less_imp_less
tff(fact_3155_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),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),A3)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_3156_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_3157_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_3158_inverse__unique,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = one_one(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = B2 ) ) ) ).

% inverse_unique
tff(fact_3159_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : divide_divide(A,A3,B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),B2)) ) ).

% field_class.field_divide_inverse
tff(fact_3160_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] : divide_divide(A,A3,B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),B2)) ) ).

% divide_inverse
tff(fact_3161_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] : divide_divide(A,A3,B2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),A3) ) ).

% divide_inverse_commute
tff(fact_3162_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),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,aa(A,fun(nat,A),power_power(A),X),M2)) ) ).

% power_mult_inverse_distrib
tff(fact_3163_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M2)) ) ).

% power_mult_power_inverse_commute
tff(fact_3164_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa2: 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),Xa2))),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),Xa2))) ) ).

% mult_inverse_of_nat_commute
tff(fact_3165_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A3)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_3166_divide__real__def,axiom,
    ! [X: real,Y: 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_3167_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_3168_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_3169_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% le_imp_inverse_le
tff(fact_3170_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% inverse_le_imp_le
tff(fact_3171_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% inverse_le_1_iff
tff(fact_3172_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).

% one_less_inverse_iff
tff(fact_3173_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% one_less_inverse
tff(fact_3174_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),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),A3),B2)),aa(A,A,inverse_inverse(A),A3))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% inverse_add
tff(fact_3175_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),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),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_3176_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).

% field_class.field_inverse
tff(fact_3177_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A3)),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),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_3178_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = divide_divide(A,one_one(A),A3) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_3179_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A3: real,C3: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),C3)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),B2),C3))) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_3180_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),B2),X))) ) ) ) ).

% scaleR_right_mono
tff(fact_3181_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),A3)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),B2)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
            & ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C3),zero_zero(real)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_3182_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C3),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),A3)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_3183_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),A3)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_3184_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,Y: A,A3: real] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),Y))) ) ) ) ).

% scaleR_left_mono
tff(fact_3185_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A3: A,C3: real] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C3),zero_zero(real)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),A3)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C3),B2))) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_3186_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,U: real,V3: real,A3: A] :
          ( ( X = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),divide_divide(real,U,V3)),A3) )
        <=> ( ( ( V3 = zero_zero(real) )
             => ( X = zero_zero(A) ) )
            & ( ( V3 != zero_zero(real) )
             => ( aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),V3),X) = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),U),A3) ) ) ) ) ) ).

% eq_vector_fraction_iff
tff(fact_3187_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V3: real,A3: A,X: A] :
          ( ( aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),divide_divide(real,U,V3)),A3) = X )
        <=> ( ( ( V3 = zero_zero(real) )
             => ( X = zero_zero(A) ) )
            & ( ( V3 != zero_zero(real) )
             => ( aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),U),A3) = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),V3),X) ) ) ) ) ) ).

% vector_fraction_eq_iff
tff(fact_3188_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,E3: A,C3: A,B2: real,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3)),E3)),D3))) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_3189_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,E3: A,C3: A,B2: real,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),B2),E3)),D3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),B2)),E3)),C3)),D3)) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_3190_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa(num,nat,numeral_numeral(nat),bit0(N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(N))) ).

% eval_nat_numeral(2)
tff(fact_3191_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y: A,N: 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(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),N)) = 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_hm(A,fun(A,fun(nat,fun(nat,A))),X),Y),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ) ).

% exp_series_add_commuting
tff(fact_3192_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : exp(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_hn(A,fun(nat,A),X))) ) ).

% exp_first_term
tff(fact_3193_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ).

% inverse_le_iff
tff(fact_3194_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ) ).

% inverse_less_iff
tff(fact_3195_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ) ).

% one_le_inverse_iff
tff(fact_3196_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).

% inverse_less_1_iff
tff(fact_3197_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).

% one_le_inverse
tff(fact_3198_BitM__plus__one,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(N)),one2) = bit0(N) ).

% BitM_plus_one
tff(fact_3199_one__plus__BitM,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(N)) = bit0(N) ).

% one_plus_BitM
tff(fact_3200_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A3)),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),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))),aa(A,A,inverse_inverse(A),B2))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_3201_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N3)))),X)) ) ) ).

% reals_Archimedean
tff(fact_3202_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2))) ) ) ).

% of_int_nonneg
tff(fact_3203_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% of_int_leD
tff(fact_3204_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2))) ) ) ).

% of_int_pos
tff(fact_3205_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),B2)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_3206_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),B2)),zero_zero(A)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_3207_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),B2))) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_3208_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X)),zero_zero(A))) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_3209_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X)),zero_zero(A))) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_3210_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X))) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_3211_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: A] :
          ( ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),B2))) ) ) ).

% split_scaleR_pos_le
tff(fact_3212_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,X: A] :
          ( ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X)),zero_zero(A))) ) ) ).

% split_scaleR_neg_le
tff(fact_3213_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: real,C3: A,D3: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),D3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),C3)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),B2),D3))) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_3214_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B2: real,X: A,Y: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X)),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),B2),Y))) ) ) ) ) ) ).

% scaleR_mono
tff(fact_3215_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,A3: real] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),one_one(real)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),A3),X)),X)) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_3216_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [X3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X3)),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),one_one(int)))))
          & ! [Y4: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y4)),X))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y4),one_one(int))))) )
             => ( Y4 = X3 ) ) ) ) ).

% floor_exists1
tff(fact_3217_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int))))) ) ) ).

% floor_exists
tff(fact_3218_forall__pos__mono__1,axiom,
    ! [P2: fun(real,bool),E3: real] :
      ( ! [D2: real,E2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),D2),E2))
         => ( pp(aa(real,bool,P2,D2))
           => pp(aa(real,bool,P2,E2)) ) )
     => ( ! [N3: nat] : pp(aa(real,bool,P2,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N3)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
         => pp(aa(real,bool,P2,E3)) ) ) ) ).

% forall_pos_mono_1
tff(fact_3219_forall__pos__mono,axiom,
    ! [P2: fun(real,bool),E3: real] :
      ( ! [D2: real,E2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),D2),E2))
         => ( pp(aa(real,bool,P2,D2))
           => pp(aa(real,bool,P2,E2)) ) )
     => ( ! [N3: nat] :
            ( ( N3 != zero_zero(nat) )
           => pp(aa(real,bool,P2,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N3)))) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
         => pp(aa(real,bool,P2,E3)) ) ) ) ).

% forall_pos_mono
tff(fact_3220_real__arch__inverse,axiom,
    ! [E3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
    <=> ? [N5: nat] :
          ( ( N5 != zero_zero(nat) )
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N5))))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N5))),E3)) ) ) ).

% real_arch_inverse
tff(fact_3221_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ? [I: int] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I)),pi) ) ).

% sin_zero_iff_int2
tff(fact_3222_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : summable(A,aTP_Lamp_ho(A,fun(nat,A),X)) ) ).

% summable_exp
tff(fact_3223_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bitM(N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),bit0(N))),one_one(A)) ) ).

% numeral_BitM
tff(fact_3224_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N3))),X)) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_3225_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),M2)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_3226_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ? [X4: 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),X4)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_3227_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),cos(A,X))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ) ).

% tan_sec
tff(fact_3228_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( cos(real,X) = zero_zero(real) )
    <=> ? [I: int] :
          ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),I))
          & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_3229_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ? [I: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),I))
          & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_3230_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_hq(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_3231_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: int] :
          ( ( ( K2 = zero_zero(int) )
           => ( aa(int,A,ring_1_of_int(A),K2) = zero_zero(A) ) )
          & ( ( K2 != zero_zero(int) )
           => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
               => ( aa(int,A,ring_1_of_int(A),K2) = aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),K2))) ) )
              & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
               => ( aa(int,A,ring_1_of_int(A),K2) = if(A,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2))))),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(int,A,ring_1_of_int(A),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))))),one_one(A))) ) ) ) ) ) ) ).

% of_int_code_if
tff(fact_3232_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))))
           => ( archimedean_round(A,X) = Y ) ) ) ) ).

% round_unique
tff(fact_3233_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),N)))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))))
         => ( archimedean_round(A,X) = N ) ) ) ).

% round_unique'
tff(fact_3234_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),X))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% of_int_round_abs_le
tff(fact_3235_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_hr(A,fun(nat,A),X),sinh(A,X)) ) ).

% sinh_converges
tff(fact_3236_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_3237_round__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: int] : archimedean_round(A,aa(int,A,ring_1_of_int(A),N)) = N ) ).

% round_of_int
tff(fact_3238_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_3239_round__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: num] : archimedean_round(A,aa(num,A,numeral_numeral(A),N)) = aa(num,int,numeral_numeral(int),N) ) ).

% round_numeral
tff(fact_3240_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_3241_round__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: nat] : archimedean_round(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% round_of_nat
tff(fact_3242_round__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: num] : archimedean_round(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)) ) ).

% round_neg_numeral
tff(fact_3243_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_round(A,Y))) ) ) ).

% round_mono
tff(fact_3244_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: A,M2: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z2),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z2))))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z2),aa(int,A,ring_1_of_int(A),M2))))) ) ).

% round_diff_minimal
tff(fact_3245_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sinh(A,X) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),exp(A,X)),aa(set(A),set(A),insert(A,one_one(A)),aa(set(A),set(A),insert(A,aa(A,A,uminus_uminus(A),one_one(A))),bot_bot(set(A)))))) ) ) ).

% sinh_zero_iff
tff(fact_3246_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% of_int_round_le
tff(fact_3247_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_ge
tff(fact_3248_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_gt
tff(fact_3249_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_hs(A,fun(nat,A),X),cosh(A,X)) ) ).

% cosh_converges
tff(fact_3250_complex__unimodular__polar,axiom,
    ! [Z2: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z2) = one_one(real) )
     => ~ ! [T6: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T6))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))
             => ( Z2 != complex2(cos(real,T6),sin(real,T6)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_3251_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K: int] : arccos(cos(real,Theta)) != aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(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),K)),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_3252_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M2),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_ht(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M2,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_3253_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,log2(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),divide_divide(real,ln_ln(real,exp(real,one_one(real))),ln_ln(real,aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))))),ln_ln(real,X)) ) ) ).

% log_base_10_eq1
tff(fact_3254_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F3,A3),B2) ).

% case_prod_conv
tff(fact_3255_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_3256_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] : unique8689654367752047608divmod(A,bit0(M2),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_hu(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M2,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_3257_complex__scaleR,axiom,
    ! [R: real,A3: real,B2: real] : aa(complex,complex,aa(real,fun(complex,complex),real_V8093663219630862766scaleR(complex),R),complex2(A3,B2)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R),A3),aa(real,real,aa(real,fun(real,real),times_times(real),R),B2)) ).

% complex_scaleR
tff(fact_3258_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q2: product_prod(A,B),F3: fun(A,fun(B,C)),G3: fun(A,fun(B,C)),P: product_prod(A,B)] :
      ( ! [X3: A,Y3: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) = Q2 )
         => ( aa(B,C,aa(A,fun(B,C),F3,X3),Y3) = aa(B,C,aa(A,fun(B,C),G3,X3),Y3) ) )
     => ( ( P = Q2 )
       => ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),P) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G3),Q2) ) ) ) ).

% split_cong
tff(fact_3259_old_Oprod_Ocase,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,fun(B,C)),X1: A,X2: B] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2)) = aa(B,C,aa(A,fun(B,C),F3,X1),X2) ).

% old.prod.case
tff(fact_3260_prod_Ocase__distrib,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,H: fun(C,D),F3: fun(A,fun(B,C)),Prod: product_prod(A,B)] : aa(C,D,H,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),Prod)) = aa(product_prod(A,B),D,aa(fun(A,fun(B,D)),fun(product_prod(A,B),D),product_case_prod(A,B,D),aa(fun(A,fun(B,C)),fun(A,fun(B,D)),aTP_Lamp_hv(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D))),H),F3)),Prod) ).

% prod.case_distrib
tff(fact_3261_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),G3: fun(product_prod(A,B),C)] :
      ( ! [X3: A,Y3: B] : aa(B,C,aa(A,fun(B,C),F3,X3),Y3) = aa(product_prod(A,B),C,G3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3) = G3 ) ) ).

% cond_case_prod_eta
tff(fact_3262_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_hw(fun(product_prod(A,B),C),fun(A,fun(B,C)),F3)) = F3 ).

% case_prod_eta
tff(fact_3263_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: fun(A,bool),P2: fun(B,fun(C,A)),Z2: product_prod(B,C)] :
      ( pp(aa(A,bool,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P2),Z2)))
     => ~ ! [X3: B,Y3: C] :
            ( ( Z2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y3) )
           => ~ pp(aa(A,bool,Q,aa(C,A,aa(B,fun(C,A),P2,X3),Y3))) ) ) ).

% case_prodE2
tff(fact_3264_internal__case__prod__def,axiom,
    ! [C: $tType,B: $tType,A: $tType] : produc5280177257484947105e_prod(A,B,C) = product_case_prod(A,B,C) ).

% internal_case_prod_def
tff(fact_3265_complex__mult,axiom,
    ! [A3: real,B2: real,C3: real,D3: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A3,B2)),complex2(C3,D3)) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),A3),C3)),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),A3),D3)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),C3))) ).

% complex_mult
tff(fact_3266_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sinh(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),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),sinh(A,Y))) ) ).

% sinh_add
tff(fact_3267_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cosh(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),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),sinh(A,Y))) ) ).

% cosh_add
tff(fact_3268_sinh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sinh(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),sinh(A,Y))) ) ).

% sinh_diff
tff(fact_3269_cosh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cosh(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),sinh(A,Y))) ) ).

% cosh_diff
tff(fact_3270_log__mult,axiom,
    ! [A3: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( ( A3 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
           => ( aa(real,real,log2(A3),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,log2(A3),X)),aa(real,real,log2(A3),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_3271_log__nat__power,axiom,
    ! [X: real,B2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,log2(B2),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log2(B2),X)) ) ) ).

% log_nat_power
tff(fact_3272_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_3273_log__of__power__less,axiom,
    ! [M2: nat,B2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),M2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log2(B2),aa(nat,real,semiring_1_of_nat(real),M2))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% log_of_power_less
tff(fact_3274_log__eq__div__ln__mult__log,axiom,
    ! [A3: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
     => ( ( A3 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
         => ( ( B2 != one_one(real) )
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
             => ( aa(real,real,log2(A3),X) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,ln_ln(real,B2),ln_ln(real,A3))),aa(real,real,log2(B2),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_3275_log__of__power__le,axiom,
    ! [M2: nat,B2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),M2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),N)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log2(B2),aa(nat,real,semiring_1_of_nat(real),M2))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% log_of_power_le
tff(fact_3276_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cosh(A,X) != zero_zero(A) )
         => ( ( cosh(A,Y) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tanh(A),X)),aa(A,A,tanh(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tanh(A),X)),aa(A,A,tanh(A),Y)))) ) ) ) ) ).

% tanh_add
tff(fact_3277_le__log2__of__power,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),M2))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M2)))) ) ).

% le_log2_of_power
tff(fact_3278_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_hx(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_3279_log2__of__power__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M2))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% log2_of_power_less
tff(fact_3280_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cosh(A,X) = zero_zero(A) )
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_3281_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_hy(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_3282_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,aa(A,fun(nat,A),power_power(A),cosh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cosh_double
tff(fact_3283_log2__of__power__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M2))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% log2_of_power_le
tff(fact_3284_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,log2(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,log2(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),exp(real,one_one(real)))),ln_ln(real,X)) ) ) ).

% log_base_10_eq2
tff(fact_3285_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_hz(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_3286_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => ( ( archimedean_ceiling(real,aa(real,real,log2(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) )
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_3287_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
         => ( archimedean_ceiling(real,aa(real,real,log2(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_3288_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( archimedean_ceiling(real,aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_3289_divmod__nat__if,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( ( N = zero_zero(nat) )
          | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) )
       => ( divmod_nat(M2,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M2) ) )
      & ( ~ ( ( N = zero_zero(nat) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) )
       => ( divmod_nat(M2,N) = aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_ia(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_3290_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log2(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K2))) = aa(nat,int,semiring_1_of_nat(int),N) )
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_3291_case__prodI,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,bool)),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),F3,A3),B2))
     => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).

% case_prodI
tff(fact_3292_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P: product_prod(A,B),C3: fun(A,fun(B,bool))] :
      ( ! [A5: A,B4: B] :
          ( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) )
         => pp(aa(B,bool,aa(A,fun(B,bool),C3,A5),B4)) )
     => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),C3),P)) ) ).

% case_prodI2
tff(fact_3293_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z2: A,C3: fun(B,fun(C,set(A))),A3: B,B2: C] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),aa(C,set(A),aa(B,fun(C,set(A)),C3,A3),B2)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),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)),C3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)))) ) ).

% mem_case_prodI
tff(fact_3294_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: product_prod(A,B),Z2: C,C3: fun(A,fun(B,set(C)))] :
      ( ! [A5: A,B4: B] :
          ( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) )
         => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Z2),aa(B,set(C),aa(A,fun(B,set(C)),C3,A5),B4))) )
     => pp(aa(set(C),bool,aa(C,fun(set(C),bool),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)),C3),P))) ) ).

% mem_case_prodI2
tff(fact_3295_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: product_prod(A,B),C3: fun(A,fun(B,fun(C,bool))),X: C] :
      ( ! [A5: A,B4: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) = P )
         => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C3,A5),B4),X)) )
     => pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),C3),P),X)) ) ).

% case_prodI2'
tff(fact_3296_floor__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int] : archim6421214686448440834_floor(A,aa(int,A,ring_1_of_int(A),Z2)) = Z2 ) ).

% floor_of_int
tff(fact_3297_ceiling__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int] : archimedean_ceiling(A,aa(int,A,ring_1_of_int(A),Z2)) = Z2 ) ).

% ceiling_of_int
tff(fact_3298_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_3299_floor__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num] : archim6421214686448440834_floor(A,aa(num,A,numeral_numeral(A),V3)) = aa(num,int,numeral_numeral(int),V3) ) ).

% floor_numeral
tff(fact_3300_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_3301_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_3302_ceiling__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num] : archimedean_ceiling(A,aa(num,A,numeral_numeral(A),V3)) = aa(num,int,numeral_numeral(int),V3) ) ).

% ceiling_numeral
tff(fact_3303_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_3304_floor__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: nat] : archim6421214686448440834_floor(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% floor_of_nat
tff(fact_3305_ceiling__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: nat] : archimedean_ceiling(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% ceiling_of_nat
tff(fact_3306_floor__uminus__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),Z2))) = aa(int,int,uminus_uminus(int),Z2) ) ).

% floor_uminus_of_int
tff(fact_3307_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),Z2) ) ).

% ceiling_add_of_int
tff(fact_3308_floor__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),Z2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X)),Z2) ) ).

% floor_diff_of_int
tff(fact_3309_ceiling__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),Z2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),Z2) ) ).

% ceiling_diff_of_int
tff(fact_3310_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ) ).

% zero_le_floor
tff(fact_3311_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A))) ) ) ).

% floor_less_zero
tff(fact_3312_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V3)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),V3)),X)) ) ) ).

% numeral_le_floor
tff(fact_3313_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).

% zero_less_floor
tff(fact_3314_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).

% floor_le_zero
tff(fact_3315_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).

% ceiling_le_zero
tff(fact_3316_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V3: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),V3))) ) ) ).

% floor_less_numeral
tff(fact_3317_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).

% zero_less_ceiling
tff(fact_3318_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).

% one_le_floor
tff(fact_3319_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V3: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(num,A,numeral_numeral(A),V3))) ) ) ).

% ceiling_le_numeral
tff(fact_3320_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).

% floor_less_one
tff(fact_3321_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).

% ceiling_less_one
tff(fact_3322_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V3)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),V3)),X)) ) ) ).

% numeral_less_ceiling
tff(fact_3323_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).

% one_le_ceiling
tff(fact_3324_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3)) ) ).

% floor_neg_numeral
tff(fact_3325_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ).

% ceiling_le_one
tff(fact_3326_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ).

% one_less_ceiling
tff(fact_3327_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V3: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(num,A,numeral_numeral(A),V3))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V3)) ) ).

% ceiling_add_numeral
tff(fact_3328_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V3: num] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(num,A,numeral_numeral(A),V3))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V3)) ) ).

% floor_diff_numeral
tff(fact_3329_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3)) ) ).

% ceiling_neg_numeral
tff(fact_3330_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_3331_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X)),one_one(int)) ) ).

% floor_diff_one
tff(fact_3332_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V3: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(num,A,numeral_numeral(A),V3))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V3)) ) ).

% ceiling_diff_numeral
tff(fact_3333_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_diff_one
tff(fact_3334_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A)))) ) ) ).

% ceiling_less_zero
tff(fact_3335_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X)) ) ) ).

% zero_le_ceiling
tff(fact_3336_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V3)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V3)),one_one(A))),X)) ) ) ).

% numeral_less_floor
tff(fact_3337_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V3: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V3)),one_one(A)))) ) ) ).

% floor_le_numeral
tff(fact_3338_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) ) ) ).

% one_less_floor
tff(fact_3339_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% floor_le_one
tff(fact_3340_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V3: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V3)),one_one(A)))) ) ) ).

% ceiling_less_numeral
tff(fact_3341_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V3)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V3)),one_one(A))),X)) ) ) ).

% numeral_le_ceiling
tff(fact_3342_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),X)) ) ) ).

% neg_numeral_le_floor
tff(fact_3343_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V3: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3)))) ) ) ).

% floor_less_neg_numeral
tff(fact_3344_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V3: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3)))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_3345_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),X)) ) ) ).

% neg_numeral_less_ceiling
tff(fact_3346_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),one_one(A))),X)) ) ) ).

% neg_numeral_less_floor
tff(fact_3347_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V3: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),one_one(A)))) ) ) ).

% floor_le_neg_numeral
tff(fact_3348_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V3: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),one_one(A)))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_3349_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),one_one(A))),X)) ) ) ).

% neg_numeral_le_ceiling
tff(fact_3350_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z2: A,C3: fun(B,fun(C,set(A))),P: product_prod(B,C)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),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)),C3),P)))
     => ~ ! [X3: B,Y3: C] :
            ( ( P = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y3) )
           => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),aa(C,set(A),aa(B,fun(C,set(A)),C3,X3),Y3))) ) ) ).

% mem_case_prodE
tff(fact_3351_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_ceiling(A,X))) ) ).

% floor_le_ceiling
tff(fact_3352_ceiling__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,X) = aa(int,int,uminus_uminus(int),archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),X))) ) ).

% ceiling_def
tff(fact_3353_floor__minus,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),X)) = aa(int,int,uminus_uminus(int),archimedean_ceiling(A,X)) ) ).

% floor_minus
tff(fact_3354_ceiling__minus,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),X)) = aa(int,int,uminus_uminus(int),archim6421214686448440834_floor(A,X)) ) ).

% ceiling_minus
tff(fact_3355_divide__complex__def,axiom,
    ! [X: complex,Y: complex] : divide_divide(complex,X,Y) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),aa(complex,complex,inverse_inverse(complex),Y)) ).

% divide_complex_def
tff(fact_3356_case__prodD,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,bool)),A3: A,B2: B] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
     => pp(aa(B,bool,aa(A,fun(B,bool),F3,A3),B2)) ) ).

% case_prodD
tff(fact_3357_case__prodE,axiom,
    ! [A: $tType,B: $tType,C3: fun(A,fun(B,bool)),P: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),C3),P))
     => ~ ! [X3: A,Y3: B] :
            ( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) )
           => ~ pp(aa(B,bool,aa(A,fun(B,bool),C3,X3),Y3)) ) ) ).

% case_prodE
tff(fact_3358_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( ( X = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
           => ( archimedean_ceiling(A,X) = archim6421214686448440834_floor(A,X) ) )
          & ( ( X != aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
           => ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) ) ) ) ) ).

% ceiling_altdef
tff(fact_3359_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),archim6421214686448440834_floor(A,X))),one_one(int))) ) ).

% ceiling_diff_floor_le_1
tff(fact_3360_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: fun(A,fun(B,fun(C,bool))),A3: A,B2: B,C3: C] :
      ( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),C3))
     => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R2,A3),B2),C3)) ) ).

% case_prodD'
tff(fact_3361_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: fun(A,fun(B,fun(C,bool))),P: product_prod(A,B),Z2: C] :
      ( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),C3),P),Z2))
     => ~ ! [X3: A,Y3: B] :
            ( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) )
           => ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C3,X3),Y3),Z2)) ) ) ).

% case_prodE'
tff(fact_3362_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))) ) ) ).

% floor_mono
tff(fact_3363_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X)) ) ).

% of_int_floor_le
tff(fact_3364_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% floor_less_cancel
tff(fact_3365_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,X))) ) ) ).

% ceiling_mono
tff(fact_3366_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ).

% le_of_int_ceiling
tff(fact_3367_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% ceiling_less_cancel
tff(fact_3368_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_round(A,X))) ) ).

% floor_le_round
tff(fact_3369_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_ceiling(A,X))) ) ).

% ceiling_ge_round
tff(fact_3370_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ib(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_id(fun(nat,fun(nat,A)),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% sum.triangle_reindex_eq
tff(fact_3371_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ib(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_if(fun(nat,fun(nat,A)),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% prod.triangle_reindex_eq
tff(fact_3372_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X)) ) ) ).

% le_floor_iff
tff(fact_3373_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),Z2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z2))) ) ) ).

% floor_less_iff
tff(fact_3374_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : pp(aa(int,bool,aa(int,fun(int,bool),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_3375_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),Z2) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z2))) ) ).

% floor_add_int
tff(fact_3376_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),archim6421214686448440834_floor(A,X)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),X)) ) ).

% int_add_floor
tff(fact_3377_floor__divide__of__int__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [K2: int,L: int] : archim6421214686448440834_floor(A,divide_divide(A,aa(int,A,ring_1_of_int(A),K2),aa(int,A,ring_1_of_int(A),L))) = divide_divide(int,K2,L) ) ).

% floor_divide_of_int_eq
tff(fact_3378_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A3)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),A3)) ) ) ).

% ceiling_le
tff(fact_3379_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),Z2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2))) ) ) ).

% ceiling_le_iff
tff(fact_3380_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),X)) ) ) ).

% less_ceiling_iff
tff(fact_3381_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : pp(aa(int,bool,aa(int,fun(int,bool),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_3382_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ig(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_id(fun(nat,fun(nat,A)),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.triangle_reindex
tff(fact_3383_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ig(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_if(fun(nat,fun(nat,A)),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.triangle_reindex
tff(fact_3384_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) ) ).

% one_add_floor
tff(fact_3385_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [M2: nat,N: nat] : archim6421214686448440834_floor(A,divide_divide(A,aa(nat,A,semiring_1_of_nat(A),M2),aa(nat,A,semiring_1_of_nat(A),N))) = aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,M2,N)) ) ).

% floor_divide_of_nat_eq
tff(fact_3386_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R),one_one(A)))) ) ).

% of_int_ceiling_le_add_one
tff(fact_3387_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R))),one_one(A))),R)) ) ).

% of_int_ceiling_diff_one_le
tff(fact_3388_prod__encode__def,axiom,
    nat_prod_encode = aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),aTP_Lamp_ih(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_3389_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P2: fun(int,bool),T2: A] :
          ( pp(aa(int,bool,P2,archim6421214686448440834_floor(A,T2)))
        <=> ! [I: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I)),T2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I)),one_one(A)))) )
             => pp(aa(int,bool,P2,I)) ) ) ) ).

% floor_split
tff(fact_3390_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( ( archim6421214686448440834_floor(A,X) = A3 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A)))) ) ) ) ).

% floor_eq_iff
tff(fact_3391_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))))
           => ( archim6421214686448440834_floor(A,X) = Z2 ) ) ) ) ).

% floor_unique
tff(fact_3392_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ).

% le_mult_floor
tff(fact_3393_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X)) ) ) ).

% less_floor_iff
tff(fact_3394_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))) ) ) ).

% floor_le_iff
tff(fact_3395_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int))))) ) ) ).

% floor_correct
tff(fact_3396_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P2: fun(int,bool),T2: A] :
          ( pp(aa(int,bool,P2,archimedean_ceiling(A,T2)))
        <=> ! [I: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I)),one_one(A))),T2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),aa(int,A,ring_1_of_int(A),I))) )
             => pp(aa(int,bool,P2,I)) ) ) ) ).

% ceiling_split
tff(fact_3397_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( ( archimedean_ceiling(A,X) = A3 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A))),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A3))) ) ) ) ).

% ceiling_eq_iff
tff(fact_3398_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2)))
           => ( archimedean_ceiling(A,X) = Z2 ) ) ) ) ).

% ceiling_unique
tff(fact_3399_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))),one_one(A))),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ) ).

% ceiling_correct
tff(fact_3400_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le
tff(fact_3401_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z2: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),Z2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))) ) ) ).

% ceiling_less_iff
tff(fact_3402_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z2: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X)) ) ) ).

% le_ceiling_iff
tff(fact_3403_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P,Q2)))),Q2)),P)) ) ) ).

% floor_divide_lower
tff(fact_3404_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_ii(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_3405_lenlex__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lenlex(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_ij(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).

% lenlex_conv
tff(fact_3406_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P,Q2)))),Q2))) ) ) ).

% ceiling_divide_upper
tff(fact_3407_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),P),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P,Q2)))),one_one(A))),Q2))) ) ) ).

% floor_divide_upper
tff(fact_3408_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),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% round_def
tff(fact_3409_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P,Q2)))),one_one(A))),Q2)),P)) ) ) ).

% ceiling_divide_lower
tff(fact_3410_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),N)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),N)),one_one(A))))
           => ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_3411_divmod__nat__def,axiom,
    ! [M2: nat,N: nat] : divmod_nat(M2,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),divide_divide(nat,M2,N)),modulo_modulo(nat,M2,N)) ).

% divmod_nat_def
tff(fact_3412_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( archim6421214686448440834_floor(real,aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log2(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_3413_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2))
         => ( archim6421214686448440834_floor(real,aa(real,real,log2(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K2))) = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_3414_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,semiring_1_of_nat(A),N) = aa(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_ik(nat,fun(nat,A))),divmod_nat(N,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ) ).

% of_nat_code_if
tff(fact_3415_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archimedean_ceiling(A,X) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archim6421214686448440834_floor(A,X) ) ) ) ) ).

% round_altdef
tff(fact_3416_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_3417_length__subseqs,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),subseqs(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_subseqs
tff(fact_3418_modulo__int__unfold,axiom,
    ! [L: int,K2: int,N: nat,M2: nat] :
      ( ( ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
          | ( aa(int,int,sgn_sgn(int),K2) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K2)),aa(nat,int,semiring_1_of_nat(int),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K2)),aa(nat,int,semiring_1_of_nat(int),M2)) ) )
      & ( ~ ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
            | ( aa(int,int,sgn_sgn(int),K2) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( aa(int,int,sgn_sgn(int),K2) = aa(int,int,sgn_sgn(int),L) )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K2)),aa(nat,int,semiring_1_of_nat(int),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M2,N))) ) )
          & ( ( aa(int,int,sgn_sgn(int),K2) != aa(int,int,sgn_sgn(int),L) )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K2)),aa(nat,int,semiring_1_of_nat(int),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M2)))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M2,N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_3419_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),A3)) = aa(A,A,sgn_sgn(A),A3) ) ).

% sgn_sgn
tff(fact_3420_frac__frac,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,archimedean_frac(A,X)) = archimedean_frac(A,X) ) ).

% frac_frac
tff(fact_3421_split__part,axiom,
    ! [B: $tType,A: $tType,P2: bool,Q: fun(A,fun(B,bool)),X5: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_il(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P2),Q)),X5))
    <=> ( pp(P2)
        & pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Q),X5)) ) ) ).

% split_part
tff(fact_3422_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_3423_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_3424_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_3425_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),A3)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_3426_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_3427_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% sgn_greater
tff(fact_3428_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,sgn_sgn(A),A3)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% sgn_less
tff(fact_3429_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A] : divide_divide(A,A3,aa(A,A,sgn_sgn(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,sgn_sgn(A),B2)) ) ).

% divide_sgn
tff(fact_3430_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_3431_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( aa(A,A,sgn_sgn(A),A3) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_3432_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_3433_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),A3)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).

% sgn_mult_self_eq
tff(fact_3434_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),A3)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_3435_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).

% sgn_abs
tff(fact_3436_dvd__mult__sgn__iff,axiom,
    ! [L: int,K2: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(int,int,sgn_sgn(int),R))))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K2))
        | ( R = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_3437_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),K2)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K2))
        | ( R = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_3438_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R))),K2))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K2))
        & ( ( R = zero_zero(int) )
         => ( K2 = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_3439_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),L)),K2))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K2))
        & ( ( R = zero_zero(int) )
         => ( K2 = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_3440_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_3441_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% sgn_of_nat
tff(fact_3442_cot__npi,axiom,
    ! [N: 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),N)),pi)) = zero_zero(real) ).

% cot_npi
tff(fact_3443_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A6: fun(A,fun(B,bool)),B6: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),A6),B6))
     => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A6))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),B6)))) ) ).

% Collect_case_prod_mono
tff(fact_3444_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_im(A,fun(B,bool))),Prod)) ).

% prod.disc_eq_case
tff(fact_3445_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,sgn_sgn(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),aa(A,A,sgn_sgn(A),X)),aa(A,A,sgn_sgn(A),Y)) ) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_3446_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B2: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),B2)) ) ).

% sgn_mult
tff(fact_3447_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A3) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,sgn_sgn(A),A3) ) ) ) ).

% same_sgn_sgn_add
tff(fact_3448_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_3449_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_3450_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_3451_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) != aa(A,A,sgn_sgn(A),A3) )
         => ( ( aa(A,A,sgn_sgn(A),A3) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B2) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_3452_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_3453_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K2: A] : aa(A,A,abs_abs(A),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),K2),aa(A,A,sgn_sgn(A),K2)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_3454_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,sgn_sgn(A),A3)) = A3 ) ).

% abs_mult_sgn
tff(fact_3455_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,abs_abs(A),A3)) = A3 ) ).

% sgn_mult_abs
tff(fact_3456_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_3457_int__sgnE,axiom,
    ! [K2: int] :
      ~ ! [N3: nat,L3: int] : K2 != aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L3)),aa(nat,int,semiring_1_of_nat(int),N3)) ).

% int_sgnE
tff(fact_3458_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A3) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_3459_div__eq__sgn__abs,axiom,
    ! [K2: int,L: int] :
      ( ( aa(int,int,sgn_sgn(int),K2) = aa(int,int,sgn_sgn(int),L) )
     => ( divide_divide(int,K2,L) = divide_divide(int,aa(int,int,abs_abs(int),K2),aa(int,int,abs_abs(int),L)) ) ) ).

% div_eq_sgn_abs
tff(fact_3460_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,X))) ) ).

% frac_ge_0
tff(fact_3461_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),archimedean_frac(A,X)),one_one(A))) ) ).

% frac_lt_1
tff(fact_3462_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = archimedean_frac(A,X) ) ).

% frac_1_eq
tff(fact_3463_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = one_one(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% sgn_1_pos
tff(fact_3464_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( ( A3 = zero_zero(A) )
           => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = zero_zero(A) ) )
          & ( ( A3 != zero_zero(A) )
           => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ) ).

% abs_sgn_eq
tff(fact_3465_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( aa(A,A,sgn_sgn(A),X) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
               => ( aa(A,A,sgn_sgn(A),X) = one_one(A) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
               => ( aa(A,A,sgn_sgn(A),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ) ) ).

% sgn_if
tff(fact_3466_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).

% sgn_1_neg
tff(fact_3467_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( real_V7770717601297561774m_norm(A,aa(A,A,sgn_sgn(A),X)) = zero_zero(real) ) )
          & ( ( X != zero_zero(A) )
           => ( real_V7770717601297561774m_norm(A,aa(A,A,sgn_sgn(A),X)) = one_one(real) ) ) ) ) ).

% norm_sgn
tff(fact_3468_div__sgn__abs__cancel,axiom,
    ! [V3: int,K2: int,L: int] :
      ( ( V3 != zero_zero(int) )
     => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V3)),aa(int,int,abs_abs(int),K2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V3)),aa(int,int,abs_abs(int),L))) = divide_divide(int,aa(int,int,abs_abs(int),K2),aa(int,int,abs_abs(int),L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_3469_div__dvd__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K2))
     => ( divide_divide(int,K2,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),K2)),aa(int,int,sgn_sgn(int),L))),divide_divide(int,aa(int,int,abs_abs(int),K2),aa(int,int,abs_abs(int),L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_3470_frac__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))) ) ).

% frac_def
tff(fact_3471_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = X )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).

% frac_eq
tff(fact_3472_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)) ) ) ) ) ).

% frac_add
tff(fact_3473_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K2: int,Q2: int] :
      ( ( aa(int,int,sgn_sgn(int),R) = aa(int,int,sgn_sgn(int),L) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R)),aa(int,int,abs_abs(int),L)))
       => ( ( K2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L)),R) )
         => eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_3474_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,Q5: int] :
            ( ( A12 = K3 )
            & ( A23 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q5: int] :
            ( ( A12 = K3 )
            & ( A23 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R5) )
            & ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L4) )
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L4)))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_3475_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) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int)) )
             => ( ( A23 != zero_zero(int) )
               => ( A12 != aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A23) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3) )
               => ( ( aa(int,int,sgn_sgn(int),R3) = aa(int,int,sgn_sgn(int),A23) )
                 => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),A23)))
                   => ( A12 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A23)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_3476_div__noneq__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero(int) )
     => ( ( aa(int,int,sgn_sgn(int),K2) != aa(int,int,sgn_sgn(int),L) )
       => ( divide_divide(int,K2,L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),divide_divide(int,aa(int,int,abs_abs(int),K2),aa(int,int,abs_abs(int),L)))),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K2)))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_3477_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),one_one(int)) ) ) ) ) ).

% floor_add
tff(fact_3478_divide__int__unfold,axiom,
    ! [L: int,K2: int,N: nat,M2: nat] :
      ( ( ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
          | ( aa(int,int,sgn_sgn(int),K2) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K2)),aa(nat,int,semiring_1_of_nat(int),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(int) ) )
      & ( ~ ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
            | ( aa(int,int,sgn_sgn(int),K2) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( aa(int,int,sgn_sgn(int),K2) = aa(int,int,sgn_sgn(int),L) )
           => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K2)),aa(nat,int,semiring_1_of_nat(int),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,M2,N)) ) )
          & ( ( aa(int,int,sgn_sgn(int),K2) != aa(int,int,sgn_sgn(int),L) )
           => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K2)),aa(nat,int,semiring_1_of_nat(int),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,M2,N)),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M2)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_3479_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less(nat))) ).

% natLess_def
tff(fact_3480_modulo__int__def,axiom,
    ! [L: int,K2: int] :
      ( ( ( L = zero_zero(int) )
       => ( modulo_modulo(int,K2,L) = K2 ) )
      & ( ( L != zero_zero(int) )
       => ( ( ( aa(int,int,sgn_sgn(int),K2) = aa(int,int,sgn_sgn(int),L) )
           => ( modulo_modulo(int,K2,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,nat2(aa(int,int,abs_abs(int),K2)),nat2(aa(int,int,abs_abs(int),L))))) ) )
          & ( ( aa(int,int,sgn_sgn(int),K2) != aa(int,int,sgn_sgn(int),L) )
           => ( modulo_modulo(int,K2,L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K2))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(aa(int,int,abs_abs(int),K2)),nat2(aa(int,int,abs_abs(int),L)))))) ) ) ) ) ) ).

% modulo_int_def
tff(fact_3481_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N)) = 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(N)))) ) ).

% mask_numeral
tff(fact_3482_num_Osize__gen_I3_J,axiom,
    ! [X33: num] : size_num(aa(num,num,bit1,X33)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X33)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(3)
tff(fact_3483_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% take_bit_rec
tff(fact_3484_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% mask_nat_positive_iff
tff(fact_3485_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_3486_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,zero_zero(nat)),A3) = zero_zero(A) ) ).

% take_bit_0
tff(fact_3487_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_3488_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_3489_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( ( bit_se2239418461657761734s_mask(A,N) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_3490_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_3491_nat__1,axiom,
    nat2(one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_3492_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_3493_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( nat2(I2) = zero_zero(nat) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),zero_zero(int))) ) ).

% nat_0_iff
tff(fact_3494_nat__le__0,axiom,
    ! [Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),zero_zero(int)))
     => ( nat2(Z2) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_3495_nat__neg__numeral,axiom,
    ! [K2: num] : nat2(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2))) = zero_zero(nat) ).

% nat_neg_numeral
tff(fact_3496_nat__zminus__int,axiom,
    ! [N: nat] : nat2(aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_3497_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ).

% take_bit_of_Suc_0
tff(fact_3498_zero__less__nat__eq,axiom,
    ! [Z2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),nat2(Z2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2)) ) ).

% zero_less_nat_eq
tff(fact_3499_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% take_bit_of_1
tff(fact_3500_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or1337092689740270186AtMost(int,L,U)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)),one_one(int))) ).

% card_atLeastAtMost_int
tff(fact_3501_nat__ceiling__le__eq,axiom,
    ! [X: real,A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),nat2(archimedean_ceiling(real,X))),A3))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),A3))) ) ).

% nat_ceiling_le_eq
tff(fact_3502_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)))
        <=> ( ( N = zero_zero(nat) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ) ).

% even_take_bit_eq
tff(fact_3503_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),nat2(Z2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z2)) ) ).

% one_less_nat_eq
tff(fact_3504_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat))),A3) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_0
tff(fact_3505_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A3: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)),nat2(A3)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_3506_nat__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,X: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),nat2(A3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_3507_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: nat,N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% take_bit_of_exp
tff(fact_3508_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_of_2
tff(fact_3509_take__bit__tightened__less__eq__nat,axiom,
    ! [M2: nat,N: nat,Q2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M2),Q2)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),Q2))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_3510_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M2)),M2)) ).

% take_bit_nat_less_eq_self
tff(fact_3511_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B2: A,M2: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M2),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,M2),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_3512_take__bit__mult,axiom,
    ! [N: nat,K2: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),K2),L)) ).

% take_bit_mult
tff(fact_3513_less__eq__mask,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),bit_se2239418461657761734s_mask(nat,N))) ).

% less_eq_mask
tff(fact_3514_nat__zero__as__int,axiom,
    zero_zero(nat) = nat2(zero_zero(int)) ).

% nat_zero_as_int
tff(fact_3515_take__bit__tightened__less__eq__int,axiom,
    ! [M2: nat,N: nat,K2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,M2),K2)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_3516_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A,B2: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = aa(A,A,bit_ri4674362597316999326ke_bit(A,N),B2) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),B2) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_3517_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Y))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),nat2(X)),nat2(Y))) ) ).

% nat_mono
tff(fact_3518_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M2: nat,N: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M2),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) = aa(A,A,if(fun(A,A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2),bit_se2584673776208193580ke_bit(A,N),bit_ri4674362597316999326ke_bit(A,M2)),A3) ) ).

% signed_take_bit_take_bit
tff(fact_3519_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M2: nat,A3: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se2638667681897837118et_bit(A,M2,A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se2638667681897837118et_bit(A,M2,A3)) = bit_se2638667681897837118et_bit(A,M2,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) ) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_3520_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M2: nat,A3: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se5668285175392031749et_bit(A,M2,A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se5668285175392031749et_bit(A,M2,A3)) = bit_se5668285175392031749et_bit(A,M2,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) ) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_3521_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M2: nat,A3: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se8732182000553998342ip_bit(A,M2,A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se8732182000553998342ip_bit(A,M2,A3)) = bit_se8732182000553998342ip_bit(A,M2,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) ) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_3522_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M2: nat,N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M2),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M2),A3) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_3523_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R),aa(nat,A,semiring_1_of_nat(A),nat2(archimedean_ceiling(A,R))))) ) ).

% of_nat_ceiling
tff(fact_3524_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),nat2(X)),N))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,semiring_1_of_nat(int),N))) ) ).

% nat_le_iff
tff(fact_3525_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z2: int] : 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),nat2(aa(int,int,abs_abs(int),W2))),nat2(aa(int,int,abs_abs(int),Z2))) ).

% nat_abs_mult_distrib
tff(fact_3526_nat__times__as__int,axiom,
    ! [X5: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X5),Xa) = nat2(aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X5)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_times_as_int
tff(fact_3527_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),bit_se2239418461657761734s_mask(nat,N))) ) ).

% less_mask
tff(fact_3528_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),nat2(archim6421214686448440834_floor(A,R)))),R)) ) ) ).

% of_nat_floor
tff(fact_3529_nat__le__eq__zle,axiom,
    ! [W2: int,Z2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),W2))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),nat2(W2)),nat2(Z2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z2)) ) ) ).

% nat_le_eq_zle
tff(fact_3530_nat__eq__iff2,axiom,
    ! [M2: nat,W2: int] :
      ( ( M2 = nat2(W2) )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( W2 = aa(nat,int,semiring_1_of_nat(int),M2) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( M2 = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff2
tff(fact_3531_nat__eq__iff,axiom,
    ! [W2: int,M2: nat] :
      ( ( nat2(W2) = M2 )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( W2 = aa(nat,int,semiring_1_of_nat(int),M2) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
         => ( M2 = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff
tff(fact_3532_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(archim6421214686448440834_floor(A,A3))),nat2(archim6421214686448440834_floor(A,B2)))),nat2(archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))))) ) ).

% le_mult_nat_floor
tff(fact_3533_split__nat,axiom,
    ! [P2: fun(nat,bool),I2: int] :
      ( pp(aa(nat,bool,P2,nat2(I2)))
    <=> ( ! [N5: nat] :
            ( ( I2 = aa(nat,int,semiring_1_of_nat(int),N5) )
           => pp(aa(nat,bool,P2,N5)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),zero_zero(int)))
         => pp(aa(nat,bool,P2,zero_zero(nat))) ) ) ) ).

% split_nat
tff(fact_3534_le__nat__iff,axiom,
    ! [K2: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),nat2(K2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),K2)) ) ) ).

% le_nat_iff
tff(fact_3535_nat__mult__distrib,axiom,
    ! [Z2: int,Z5: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
     => ( 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),nat2(Z2)),nat2(Z5)) ) ) ).

% nat_mult_distrib
tff(fact_3536_Suc__as__int,axiom,
    ! [X5: nat] : aa(nat,nat,suc,X5) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X5)),one_one(int))) ).

% Suc_as_int
tff(fact_3537_nat__abs__triangle__ineq,axiom,
    ! [K2: int,L: int] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),nat2(aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat2(aa(int,int,abs_abs(int),K2))),nat2(aa(int,int,abs_abs(int),L))))) ).

% nat_abs_triangle_ineq
tff(fact_3538_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( nat2(archim6421214686448440834_floor(real,X)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_3539_div__abs__eq__div__nat,axiom,
    ! [K2: int,L: int] : divide_divide(int,aa(int,int,abs_abs(int),K2),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,nat2(aa(int,int,abs_abs(int),K2)),nat2(aa(int,int,abs_abs(int),L)))) ).

% div_abs_eq_div_nat
tff(fact_3540_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( nat2(archim6421214686448440834_floor(real,X)) = N ) ) ) ).

% floor_eq3
tff(fact_3541_le__nat__floor,axiom,
    ! [X: nat,A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),X)),A3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),nat2(archim6421214686448440834_floor(real,A3)))) ) ).

% le_nat_floor
tff(fact_3542_mod__abs__eq__div__nat,axiom,
    ! [K2: int,L: int] : modulo_modulo(int,aa(int,int,abs_abs(int),K2),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(aa(int,int,abs_abs(int),K2)),nat2(aa(int,int,abs_abs(int),L)))) ).

% mod_abs_eq_div_nat
tff(fact_3543_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K2: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),bit0(K2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),K2))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_bit0
tff(fact_3544_nat__2,axiom,
    nat2(aa(num,int,numeral_numeral(int),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% nat_2
tff(fact_3545_sgn__power__injE,axiom,
    ! [A3: real,N: nat,X: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),A3)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),A3)),N)) = X )
     => ( ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),B2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),B2)),N)) )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( A3 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_3546_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
     => ( aa(nat,nat,suc,nat2(Z2)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_3547_nat__mult__distrib__neg,axiom,
    ! [Z2: int,Z5: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),zero_zero(int)))
     => ( 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),nat2(aa(int,int,uminus_uminus(int),Z2))),nat2(aa(int,int,uminus_uminus(int),Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_3548_nat__abs__int__diff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
       => ( nat2(aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),A3) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
       => ( nat2(aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2) ) ) ) ).

% nat_abs_int_diff
tff(fact_3549_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( nat2(archim6421214686448440834_floor(real,X)) = N ) ) ) ).

% floor_eq4
tff(fact_3550_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_3551_diff__nat__eq__if,axiom,
    ! [Z5: int,Z2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z5),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),nat2(Z2)),nat2(Z5)) = nat2(Z2) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z5),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),nat2(Z2)),nat2(Z5)) = if(nat,aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z2),Z5)),zero_zero(int)),zero_zero(nat),nat2(aa(int,int,aa(int,fun(int,int),minus_minus(int),Z2),Z5))) ) ) ) ).

% diff_nat_eq_if
tff(fact_3552_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = zero_zero(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),A3)) ) ) ).

% take_bit_eq_0_iff
tff(fact_3553_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K2: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),bit0(K2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K2))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_numeral_bit0
tff(fact_3554_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M2)),M2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),M2)) ) ).

% take_bit_nat_less_self_iff
tff(fact_3555_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa(nat,nat,suc,bit_se2239418461657761734s_mask(nat,N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% Suc_mask_eq_exp
tff(fact_3556_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K2: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K2)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% take_bit_Suc_minus_bit0
tff(fact_3557_nat__dvd__iff,axiom,
    ! [Z2: int,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),nat2(Z2)),M2))
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
         => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Z2),aa(nat,int,semiring_1_of_nat(int),M2))) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
         => ( M2 = zero_zero(nat) ) ) ) ) ).

% nat_dvd_iff
tff(fact_3558_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,N)))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_3559_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: 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(K2)))) = 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),K2)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% take_bit_numeral_minus_bit0
tff(fact_3560_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,suc,N))),one_one(A)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_3561_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K2: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),K2))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_3562_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A3) = 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,N),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% take_bit_Suc
tff(fact_3563_take__bit__int__less__eq,axiom,
    ! [N: nat,K2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_3564_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K2: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_3565_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( ( divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
             => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = zero_zero(A) ) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
             => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A)) ) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_3566_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K2: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K2))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_3567_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero(real) )
     => ( arctan(divide_divide(real,one_one(real),X)) = aa(real,real,aa(real,fun(real,real),minus_minus(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)))),arctan(X)) ) ) ).

% arctan_inverse
tff(fact_3568_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : size_num(bit0(X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_3569_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: 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,K2)))) = 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(K2))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_3570_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),S3)))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R2),S3)) ) ).

% pred_subset_eq2
tff(fact_3571_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K2)))) = 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,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K2))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_3572_or__int__unfold,axiom,
    ! [K2: int,L: int] :
      ( ( ( ( K2 = aa(int,int,uminus_uminus(int),one_one(int)) )
          | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) )
      & ( ~ ( ( K2 = aa(int,int,uminus_uminus(int),one_one(int)) )
            | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
       => ( ( ( K2 = zero_zero(int) )
           => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),L) = L ) )
          & ( ( K2 != zero_zero(int) )
           => ( ( ( L = zero_zero(int) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),L) = K2 ) )
              & ( ( L != zero_zero(int) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),L) = 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,K2,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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_3573_arctan__half,axiom,
    ! [X: real] : arctan(X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),arctan(divide_divide(real,X,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),sqrt(aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% arctan_half
tff(fact_3574_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A3) = A3 ) ).

% or.left_neutral
tff(fact_3575_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),zero_zero(A)) = A3 ) ).

% or.right_neutral
tff(fact_3576_real__sqrt__mult__self,axiom,
    ! [A3: real] : aa(real,real,aa(real,fun(real,real),times_times(real),sqrt(A3)),sqrt(A3)) = aa(real,real,abs_abs(real),A3) ).

% real_sqrt_mult_self
tff(fact_3577_real__sqrt__abs2,axiom,
    ! [X: 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_3578_pred__numeral__inc,axiom,
    ! [K2: num] : pred_numeral(inc(K2)) = aa(num,nat,numeral_numeral(nat),K2) ).

% pred_numeral_inc
tff(fact_3579_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_3580_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M2))) ) ).

% add_neg_numeral_special(6)
tff(fact_3581_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: 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),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% add_neg_numeral_special(5)
tff(fact_3582_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M2)) ) ).

% diff_numeral_special(6)
tff(fact_3583_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% diff_numeral_special(5)
tff(fact_3584_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : aa(nat,real,aa(real,fun(nat,real),power_power(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,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(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,aa(real,fun(nat,real),power_power(real),Xa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(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,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(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,aa(real,fun(nat,real),power_power(real),Xa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_3585_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_3586_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_3587_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_3588_real__sqrt__mult,axiom,
    ! [X: real,Y: 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),sqrt(X)),sqrt(Y)) ).

% real_sqrt_mult
tff(fact_3589_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_3590_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_3591_num__induct,axiom,
    ! [P2: fun(num,bool),X: num] :
      ( pp(aa(num,bool,P2,one2))
     => ( ! [X3: num] :
            ( pp(aa(num,bool,P2,X3))
           => pp(aa(num,bool,P2,inc(X3))) )
       => pp(aa(num,bool,P2,X)) ) ) ).

% num_induct
tff(fact_3592_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_3593_inc_Osimps_I1_J,axiom,
    inc(one2) = bit0(one2) ).

% inc.simps(1)
tff(fact_3594_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(bit0(X)) = aa(num,num,bit1,X) ).

% inc.simps(2)
tff(fact_3595_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(aa(num,num,bit1,X)) = bit0(inc(X)) ).

% inc.simps(3)
tff(fact_3596_add__One,axiom,
    ! [X: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2) = inc(X) ).

% add_One
tff(fact_3597_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),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_3598_inc__BitM__eq,axiom,
    ! [N: num] : inc(bitM(N)) = bit0(N) ).

% inc_BitM_eq
tff(fact_3599_BitM__inc__eq,axiom,
    ! [N: num] : bitM(inc(N)) = aa(num,num,bit1,N) ).

% BitM_inc_eq
tff(fact_3600_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_3601_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),X) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),X) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),Y) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( X = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_3602_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_3603_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B)),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),S3)),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R2),S3))) ) ).

% sup_Un_eq2
tff(fact_3604_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( ! [X4: A,Xa3: B] :
          ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3)),R2))
        <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa3)),S3)) )
    <=> ( R2 = S3 ) ) ).

% pred_equals_eq2
tff(fact_3605_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),bot_bot(set(product_prod(A,B))))) ) ).

% bot_empty_eq2
tff(fact_3606_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B)),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),S3)),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),R2),S3))) ) ).

% inf_Int_eq2
tff(fact_3607_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),bit_se2239418461657761734s_mask(A,N)) ) ).

% mask_Suc_exp
tff(fact_3608_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = 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,N))) ) ).

% mask_Suc_double
tff(fact_3609_subrelI,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X3: A,Y3: B] :
          ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),R))
         => pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),S2)) )
     => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R),S2)) ) ).

% subrelI
tff(fact_3610_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(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,aa(real,fun(nat,real),power_power(real),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(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,aa(real,fun(nat,real),power_power(real),Xa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_3611_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sqrt(aa(real,real,aa(real,fun(real,real),times_times(real),X),Y))),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_3612_or__int__rec,axiom,
    ! [K2: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fdisj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K2)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% or_int_rec
tff(fact_3613_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K2: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),K2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,N))),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_3614_set__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs)) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),bool),aTP_Lamp_io(nat,fun(list(A),fun(list(A),bool)),N),Xs)) ).

% set_n_lists
tff(fact_3615_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_3616_take__bit__Suc__from__most,axiom,
    ! [N: nat,K2: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),K2) = 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,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N)))),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K2)) ).

% take_bit_Suc_from_most
tff(fact_3617_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_ip(real,bool))) ).

% pi_def
tff(fact_3618_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,bool)) ) ) ).

% bit_0_eq
tff(fact_3619_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_3620_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_3621_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(M2))),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M2)),N)) ) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_3622_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M2))),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M2)),N)) ) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_3623_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_3624_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_3625_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W2)))),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W2))),N)) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_3626_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( pp(aa(nat,bool,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,N)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W2)),N)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_3627_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),zero_zero(nat)))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ).

% bit_0
tff(fact_3628_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),N))
        <=> ( ( N = zero_zero(nat) )
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ) ).

% bit_mod_2_iff
tff(fact_3629_bot2E,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : ~ pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X),Y)) ).

% bot2E
tff(fact_3630_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(nat,nat,suc,N))) ) ).

% not_bit_1_Suc
tff(fact_3631_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),N))
        <=> ( N = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_3632_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: bool,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(bool,A,zero_neq_one_of_bool(A),B2)),N))
        <=> ( pp(B2)
            & ( N = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_3633_cis__mult,axiom,
    ! [A3: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),cis(A3)),cis(B2)) = cis(aa(real,real,aa(real,fun(real,real),plus_plus(real),A3),B2)) ).

% cis_mult
tff(fact_3634_DeMoivre,axiom,
    ! [A3: real,N: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A3)),N) = cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),A3)) ).

% DeMoivre
tff(fact_3635_bit__concat__bit__iff,axiom,
    ! [M2: nat,K2: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,bit_concat_bit(M2,K2,L)),N))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N)) )
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_3636_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,A3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
         => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N)) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_3637_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),N)) ) ) ).

% bit_Suc
tff(fact_3638_length__n__lists__elem,axiom,
    ! [A: $tType,Ys: list(A),N: nat,Xs: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs))))
     => ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ).

% length_n_lists_elem
tff(fact_3639_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N3: nat] :
          ( ! [M3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),M3))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),M3))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N3)) ) )
         => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N3),one_one(nat))))
              <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N3)) ) ) ) ).

% int_bit_bound
tff(fact_3640_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = zero_zero(A) )
        <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N)) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_3641_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_3642_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_3643_set__bit__eq,axiom,
    ! [N: nat,K2: int] : bit_se5668285175392031749et_bit(int,N,K2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),aa(int,int,aa(int,fun(int,int),times_times(int),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ).

% set_bit_eq
tff(fact_3644_unset__bit__eq,axiom,
    ! [N: nat,K2: int] : bit_se2638667681897837118et_bit(int,N,K2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),aa(int,int,aa(int,fun(int,int),times_times(int),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),N))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ).

% unset_bit_eq
tff(fact_3645_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,N: nat] :
          ( ! [J3: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,suc,J3)))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),N))
          <=> ( ( ( N = zero_zero(nat) )
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) )
              & ( ( N != zero_zero(nat) )
               => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)),N)) ) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_3646_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
        <=> ( ( ( N = zero_zero(nat) )
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) )
            & ( ( N != zero_zero(nat) )
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ) ).

% bit_rec
tff(fact_3647_or__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% or_Suc_0_eq
tff(fact_3648_Suc__0__or__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% Suc_0_or_eq
tff(fact_3649_or__nat__rec,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fdisj(aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2)),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),divide_divide(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% or_nat_rec
tff(fact_3650_or__nat__unfold,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M2),N) = N ) )
      & ( ( M2 != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M2),N) = M2 ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),divide_divide(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_3651_old_Orec__prod__def,axiom,
    ! [T: $tType,B: $tType,A: $tType,X5: fun(A,fun(B,T)),Xa: product_prod(A,B)] : product_rec_prod(A,B,T,X5,Xa) = the(T,product_rec_set_prod(A,B,T,X5,Xa)) ).

% old.rec_prod_def
tff(fact_3652_length__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_n_lists
tff(fact_3653_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => bij_betw(nat,complex,aTP_Lamp_iq(nat,fun(nat,complex),N),aa(nat,set(nat),set_ord_lessThan(nat),N),aa(fun(complex,bool),set(complex),collect(complex),aTP_Lamp_an(nat,fun(complex,bool),N))) ) ).

% bij_betw_roots_unity
tff(fact_3654_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),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)),N)) = one_one(complex) ) ) ).

% cis_multiple_2pi
tff(fact_3655_frac__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),archimedean_frac(A,X)),ring_1_Ints(A)))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A))) ) ) ).

% frac_in_Ints_iff
tff(fact_3656_The__split__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_ir(A,fun(B,fun(A,fun(B,bool))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% The_split_eq
tff(fact_3657_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A))) ) ) ).

% frac_eq_0_iff
tff(fact_3658_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
            | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),ring_1_Ints(A))) )
         => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) ) ) ).

% floor_add2
tff(fact_3659_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),archimedean_frac(A,X)))
        <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A))) ) ) ).

% frac_gt_0_iff
tff(fact_3660_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),ring_1_Ints(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_mult
tff(fact_3661_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),ring_1_Ints(A))) ) ).

% Ints_0
tff(fact_3662_finite__same__card__bij,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B6))
       => ( ( aa(set(A),nat,finite_card(A),A6) = aa(set(B),nat,finite_card(B),B6) )
         => ? [H3: fun(A,B)] : bij_betw(A,B,H3,A6,B6) ) ) ) ).

% finite_same_card_bij
tff(fact_3663_bij__betw__iff__card,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B6))
       => ( ? [F5: fun(A,B)] : bij_betw(A,B,F5,A6,B6)
        <=> ( aa(set(A),nat,finite_card(A),A6) = aa(set(B),nat,finite_card(B),B6) ) ) ) ) ).

% bij_betw_iff_card
tff(fact_3664_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_3665_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),N))
    <=> ( N = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_3666_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),ring_1_Ints(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_3667_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B2: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_is(A,fun(A,fun(A,bool)),A3),B2)))) ) ).

% finite_int_segment
tff(fact_3668_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),N))) ).

% not_bit_Suc_0_numeral
tff(fact_3669_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),ring_1_Ints(A)))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_3670_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A6: set(A),A3: A] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ? [H3: fun(A,A)] : bij_betw(A,A,H3,A6,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_3671_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A6: set(A),A3: A] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ? [H3: fun(A,A)] : bij_betw(A,A,H3,A6,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_3672_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_it(A,fun(A,bool),A3)))) ) ).

% finite_abs_int_segment
tff(fact_3673_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),M5))
     => ? [H3: fun(nat,A)] : bij_betw(nat,A,H3,set_or1337092689740270186AtMost(nat,one_one(nat),aa(set(A),nat,finite_card(A),M5)),M5) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_3674_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S5: set(B),T4: set(C),H: fun(B,C),S3: set(B),T5: set(C),G3: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T4))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S5),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),S5))
                   => ( aa(C,A,G3,aa(B,C,H,A5)) = zero_zero(A) ) )
               => ( ! [B4: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B4),T4))
                     => ( aa(C,A,G3,B4) = zero_zero(A) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_iu(fun(B,C),fun(fun(C,A),fun(B,A)),H),G3)),S3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G3),T5) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_3675_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),ring_1_Ints(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3)),zero_zero(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% Ints_odd_less_0
tff(fact_3676_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( ( X != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),X))) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_3677_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A)))
           => ( X = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_3678_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) )
    <=> pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),ring_1_Ints(real))) ) ).

% sin_times_pi_eq_0
tff(fact_3679_bij__betw__nth,axiom,
    ! [A: $tType,Xs: list(A),A6: set(nat),B6: set(A)] :
      ( distinct(A,Xs)
     => ( ( A6 = aa(nat,set(nat),set_ord_lessThan(nat),aa(list(A),nat,size_size(list(A)),Xs)) )
       => ( ( B6 = aa(list(A),set(A),set2(A),Xs) )
         => bij_betw(nat,A,nth(A,Xs),A6,B6) ) ) ) ).

% bij_betw_nth
tff(fact_3680_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
           => ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = zero_zero(A) ) )
          & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
           => ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,X)) ) ) ) ) ).

% frac_neg
tff(fact_3681_le__mult__floor__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A3: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),ring_1_Ints(B)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(B,A3)),archim6421214686448440834_floor(B,B2)))),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2))))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_3682_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: A] :
          ( ( archimedean_frac(A,X) = A3 )
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3)),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) ) ) ) ).

% frac_unique_iff
tff(fact_3683_mult__ceiling__le__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A3: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),ring_1_Ints(B)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(B,A3)),archimedean_ceiling(B,B2))))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_3684_sin__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),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)),N)) = zero_zero(real) ) ) ).

% sin_integer_2pi
tff(fact_3685_cos__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),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)),N)) = one_one(real) ) ) ).

% cos_integer_2pi
tff(fact_3686_i__even__power,axiom,
    ! [N: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),imaginary_unit),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),aa(complex,complex,uminus_uminus(complex),one_one(complex))),N) ).

% i_even_power
tff(fact_3687_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ).

% Suc_0_xor_eq
tff(fact_3688_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ).

% xor_Suc_0_eq
tff(fact_3689_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list(bool)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(list(bool),int,aa(int,fun(list(bool),int),aa(fun(bool,int),fun(int,fun(list(bool),int)),groups4207007520872428315er_sum(bool,int),zero_neq_one_of_bool(int)),aa(num,int,numeral_numeral(int),bit0(one2))),Bs)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(list(bool),nat,size_size(list(bool)),Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_3690_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_3691_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),A3) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_3692_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A3) = A3 ) ).

% xor.left_neutral
tff(fact_3693_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),zero_zero(A)) = A3 ) ).

% xor.right_neutral
tff(fact_3694_complex__i__mult__minus,axiom,
    ! [X: 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),imaginary_unit),X)) = aa(complex,complex,uminus_uminus(complex),X) ).

% complex_i_mult_minus
tff(fact_3695_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A3: A,X: B,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F3,X)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs))) ) ).

% horner_sum_simps(2)
tff(fact_3696_divide__numeral__i,axiom,
    ! [Z2: complex,N: num] : divide_divide(complex,Z2,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),N)),imaginary_unit)) = 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),N)) ).

% divide_numeral_i
tff(fact_3697_divide__i,axiom,
    ! [X: complex] : divide_divide(complex,X,imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,uminus_uminus(complex),imaginary_unit)),X) ).

% divide_i
tff(fact_3698_i__squared,axiom,
    aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),imaginary_unit) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% i_squared
tff(fact_3699_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_3700_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_3701_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_3702_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_3703_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_3704_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_3705_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_3706_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_3707_i__times__eq__iff,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),W2) = Z2 )
    <=> ( W2 = aa(complex,complex,uminus_uminus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z2)) ) ) ).

% i_times_eq_iff
tff(fact_3708_i__mult__Complex,axiom,
    ! [A3: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),complex2(A3,B2)) = complex2(aa(real,real,uminus_uminus(real),B2),A3) ).

% i_mult_Complex
tff(fact_3709_Complex__mult__i,axiom,
    ! [A3: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A3,B2)),imaginary_unit) = complex2(aa(real,real,uminus_uminus(real),B2),A3) ).

% Complex_mult_i
tff(fact_3710_xor__nat__unfold,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( M2 = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M2),N) = N ) )
      & ( ( M2 != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M2),N) = M2 ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),divide_divide(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_3711_xor__nat__rec,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2))),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),divide_divide(nat,M2,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% xor_nat_rec
tff(fact_3712_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list(bool),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(list(bool),A,aa(A,fun(list(bool),A),aa(fun(bool,A),fun(A,fun(list(bool),A)),groups4207007520872428315er_sum(bool,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),bit0(one2))),Bs)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(bool),nat,size_size(list(bool)),Bs)))
            & pp(aa(nat,bool,nth(bool,Bs),N)) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_3713_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)),P) = P ).

% case_prod_Pair_iden
tff(fact_3714_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I5: set(A),F3: fun(A,B),I2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),F4))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_iv(set(A),fun(fun(A,B),fun(A,bool)),I5),F3))),F4))
           => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert(A,I2),bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F3),I5)),aa(A,B,F3,I2)) ) )
              & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert(A,I2),bot_bot(set(A))))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F3),I5) ) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3715_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite2(A),S3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),S3)))
           => ( infini527867602293511546merate(A,S3,aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,bool),aTP_Lamp_iw(set(A),fun(nat,fun(A,bool)),S3),N)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_3716_Sum__Ico__nat,axiom,
    ! [M2: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_bu(nat,nat)),set_or7035219750837199246ssThan(nat,M2,N)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Ico_nat
tff(fact_3717_Least__eq__0,axiom,
    ! [P2: fun(nat,bool)] :
      ( pp(aa(nat,bool,P2,zero_zero(nat)))
     => ( ord_Least(nat,P2) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_3718_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa(set(nat),bool,finite_finite2(nat),set_or7035219750837199246ssThan(nat,L,U))) ).

% finite_atLeastLessThan
tff(fact_3719_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),set_or7035219750837199246ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),U)) ) ) ) ).

% atLeastLessThan_iff
tff(fact_3720_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_3721_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: A,J2: A,M2: A,N: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I2,J2)),set_or7035219750837199246ssThan(A,M2,N)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J2),I2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),I2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J2),N)) ) ) ) ) ).

% ivl_subset
tff(fact_3722_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_3723_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B2) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_3724_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or7035219750837199246ssThan(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Ico_iff
tff(fact_3725_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: A,N: A,M2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),N))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or7035219750837199246ssThan(A,I2,M2)),set_or7035219750837199246ssThan(A,I2,N)) = set_or7035219750837199246ssThan(A,N,M2) ) ) ) ).

% ivl_diff
tff(fact_3726_lessThan__minus__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [N: A,M2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(A,set(A),set_ord_lessThan(A),N)),aa(A,set(A),set_ord_lessThan(A),M2)) = set_or7035219750837199246ssThan(A,M2,N) ) ).

% lessThan_minus_lessThan
tff(fact_3727_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_3728_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or7035219750837199246ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ).

% card_atLeastLessThan
tff(fact_3729_atLeastLessThan__singleton,axiom,
    ! [M2: nat] : set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,M2)) = aa(set(nat),set(nat),insert(nat,M2),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_3730_sum_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P: fun(B,A),I2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I5),P))))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P),aa(set(B),set(B),insert(B,I2),I5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P),I5) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P),aa(set(B),set(B),insert(B,I2),I5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,P,I2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P),I5)) ) ) ) ) ) ).

% sum.insert'
tff(fact_3731_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M2: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G3,N)) ) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_3732_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M2: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G3,N)) ) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_3733_LeastI2,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),A3: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P2,A3))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,P2,X3))
               => pp(aa(A,bool,Q,X3)) )
           => pp(aa(A,bool,Q,ord_Least(A,P2))) ) ) ) ).

% LeastI2
tff(fact_3734_LeastI__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool)] :
          ( ? [X_1: A] : pp(aa(A,bool,P2,X_1))
         => pp(aa(A,bool,P2,ord_Least(A,P2))) ) ) ).

% LeastI_ex
tff(fact_3735_LeastI2__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),Q: fun(A,bool)] :
          ( ? [X_1: A] : pp(aa(A,bool,P2,X_1))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,P2,X3))
               => pp(aa(A,bool,Q,X3)) )
           => pp(aa(A,bool,Q,ord_Least(A,P2))) ) ) ) ).

% LeastI2_ex
tff(fact_3736_LeastI,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),K2: A] :
          ( pp(aa(A,bool,P2,K2))
         => pp(aa(A,bool,P2,ord_Least(A,P2))) ) ) ).

% LeastI
tff(fact_3737_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
           => ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C3,D3) )
            <=> ( ( A3 = C3 )
                & ( B2 = D3 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_3738_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C3,D3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
             => ( A3 = C3 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_3739_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C3,D3) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C3),D3))
             => ( B2 = D3 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_3740_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),Q: fun(A,bool)] :
          ( ? [X_1: A] : pp(aa(A,bool,P2,X_1))
         => ( ! [A5: A] :
                ( pp(aa(A,bool,P2,A5))
               => ( ! [B7: A] :
                      ( pp(aa(A,bool,P2,B7))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),B7)) )
                 => pp(aa(A,bool,Q,A5)) ) )
           => pp(aa(A,bool,Q,ord_Least(A,P2))) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_3741_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),A3: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P2,A3))
         => ( ! [A5: A] :
                ( pp(aa(A,bool,P2,A5))
               => ( ! [B7: A] :
                      ( pp(aa(A,bool,P2,B7))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),B7)) )
                 => pp(aa(A,bool,Q,A5)) ) )
           => pp(aa(A,bool,Q,ord_Least(A,P2))) ) ) ) ).

% LeastI2_wellorder
tff(fact_3742_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P2,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P2,Y3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( ord_Least(A,P2) = X ) ) ) ) ).

% Least_equality
tff(fact_3743_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool),X: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P2,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P2,Y3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P2,X3))
                 => ( ! [Y4: A] :
                        ( pp(aa(A,bool,P2,Y4))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4)) )
                   => pp(aa(A,bool,Q,X3)) ) )
             => pp(aa(A,bool,Q,ord_Least(A,P2))) ) ) ) ) ).

% LeastI2_order
tff(fact_3744_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool),Z2: A] :
          ( ? [X5: A] :
              ( pp(aa(A,bool,P2,X5))
              & ! [Y3: A] :
                  ( pp(aa(A,bool,P2,Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y3)) )
              & ! [Y3: A] :
                  ( ( pp(aa(A,bool,P2,Y3))
                    & ! [Ya2: A] :
                        ( pp(aa(A,bool,P2,Ya2))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),Ya2)) ) )
                 => ( Y3 = X5 ) ) )
         => ( pp(aa(A,bool,P2,Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P2)),Z2)) ) ) ) ).

% Least1_le
tff(fact_3745_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool)] :
          ( ? [X5: A] :
              ( pp(aa(A,bool,P2,X5))
              & ! [Y3: A] :
                  ( pp(aa(A,bool,P2,Y3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y3)) )
              & ! [Y3: A] :
                  ( ( pp(aa(A,bool,P2,Y3))
                    & ! [Ya2: A] :
                        ( pp(aa(A,bool,P2,Ya2))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),Ya2)) ) )
                 => ( Y3 = X5 ) ) )
         => pp(aa(A,bool,P2,ord_Least(A,P2))) ) ) ).

% Least1I
tff(fact_3746_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),I5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G3),aa(fun(B,bool),set(B),collect(B),aa(set(B),fun(B,bool),aTP_Lamp_ix(fun(B,A),fun(set(B),fun(B,bool)),G3),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G3),I5) ) ).

% sum.non_neutral'
tff(fact_3747_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,bool),K2: A] :
          ( pp(aa(A,bool,P2,K2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P2)),K2)) ) ) ).

% Least_le
tff(fact_3748_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K2: A,P2: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),ord_Least(A,P2)))
         => ~ pp(aa(A,bool,P2,K2)) ) ) ).

% not_less_Least
tff(fact_3749_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C3,D3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_3750_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or7035219750837199246ssThan(A,A3,B2))) ) ) ).

% infinite_Ico
tff(fact_3751_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_3752_all__nat__less__eq,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ! [M6: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M6),N))
         => pp(aa(nat,bool,P2,M6)) )
    <=> ! [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P2,X4)) ) ) ).

% all_nat_less_eq
tff(fact_3753_ex__nat__less__eq,axiom,
    ! [N: nat,P2: fun(nat,bool)] :
      ( ? [M6: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M6),N))
          & pp(aa(nat,bool,P2,M6)) )
    <=> ? [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P2,X4)) ) ) ).

% ex_nat_less_eq
tff(fact_3754_ivl__disj__int__two_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(3)
tff(fact_3755_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,L,aa(nat,nat,suc,U)) = set_or1337092689740270186AtMost(nat,L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_3756_lessThan__atLeast0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),N) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ).

% lessThan_atLeast0
tff(fact_3757_atLeastLessThan0,axiom,
    ! [M2: nat] : set_or7035219750837199246ssThan(nat,M2,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_3758_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bk(fun(nat,A),fun(nat,A),G3)),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_3759_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M2: nat,K2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_bl(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_3760_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_da(fun(nat,A),fun(nat,A),G3)),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_3761_Least__Suc2,axiom,
    ! [P2: fun(nat,bool),N: nat,Q: fun(nat,bool),M2: nat] :
      ( pp(aa(nat,bool,P2,N))
     => ( pp(aa(nat,bool,Q,M2))
       => ( ~ pp(aa(nat,bool,P2,zero_zero(nat)))
         => ( ! [K: nat] :
                ( pp(aa(nat,bool,P2,aa(nat,nat,suc,K)))
              <=> pp(aa(nat,bool,Q,K)) )
           => ( ord_Least(nat,P2) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_3762_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M2: nat,K2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dc(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_3763_sum_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_add(A) )
     => ! [A3: B,C3: B,B2: B,D3: B,G3: fun(B,A),H: fun(B,A)] :
          ( ( A3 = C3 )
         => ( ( B2 = D3 )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C3),X3))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D3))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),set_or7035219750837199246ssThan(B,A3,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),set_or7035219750837199246ssThan(B,C3,D3)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_3764_Least__Suc,axiom,
    ! [P2: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P2,N))
     => ( ~ pp(aa(nat,bool,P2,zero_zero(nat)))
       => ( ord_Least(nat,P2) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_iy(fun(nat,bool),fun(nat,bool),P2))) ) ) ) ).

% Least_Suc
tff(fact_3765_prod_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_mult(A) )
     => ! [A3: B,C3: B,B2: B,D3: B,G3: fun(B,A),H: fun(B,A)] :
          ( ( A3 = C3 )
         => ( ( B2 = D3 )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C3),X3))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D3))
                   => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),set_or7035219750837199246ssThan(B,A3,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),set_or7035219750837199246ssThan(B,C3,D3)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_3766_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_3767_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),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_3768_ex__bij__betw__finite__nat,axiom,
    ! [A: $tType,M5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),M5))
     => ? [H3: fun(A,nat)] : bij_betw(A,nat,H3,M5,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M5))) ) ).

% ex_bij_betw_finite_nat
tff(fact_3769_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,P: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P))
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,N,P))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,P)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_3770_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,P: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P))
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,M2,P))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,M2,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,N,P)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_3771_ivl__disj__int__two_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(7)
tff(fact_3772_ivl__disj__int__one_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(2)
tff(fact_3773_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,P: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,N,P))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,P)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_3774_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_3775_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A)] : infini527867602293511546merate(A,S3,zero_zero(nat)) = ord_Least(A,aTP_Lamp_iz(set(A),fun(A,bool),S3)) ) ).

% enumerate_0
tff(fact_3776_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N6: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N6),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite2(nat),N6)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_3777_sum_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),T5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
               => ( aa(B,A,G3,X3) = zero_zero(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S3))
                 => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G3),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H),S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3778_sum_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),T5: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
               => ( aa(B,A,H,I3) = zero_zero(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S3))
                 => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G3),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H),T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3779_sum_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),T5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
               => ( aa(B,A,G3,X3) = zero_zero(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G3),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G3),S3) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3780_sum_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(B),T5: set(B),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
               => ( aa(B,A,G3,X3) = zero_zero(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G3),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G3),T5) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3781_subset__card__intvl__is__intvl,axiom,
    ! [A6: set(nat),K2: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),A6),set_or7035219750837199246ssThan(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(set(nat),nat,finite_card(nat),A6)))))
     => ( A6 = set_or7035219750837199246ssThan(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(set(nat),nat,finite_card(nat),A6))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_3782_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( set_or7035219750837199246ssThan(nat,I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I2,J2)),set_or7035219750837199246ssThan(nat,J2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_3783_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or7035219750837199246ssThan(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D3)) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_3784_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or1337092689740270186AtMost(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_3785_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_3786_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_3787_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G3,N)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_3788_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_3789_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: nat,B2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_3790_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G3,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_3791_sum_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I5),G3))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_ja(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H)),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G3),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H),I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_3792_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] : set_or7035219750837199246ssThan(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_3793_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_3794_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: nat,B2: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_3795_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P: fun(B,A)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I5),P))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P),I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I5),P))) ) )
          & ( ~ pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I5),P))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P),I5) = zero_zero(A) ) ) ) ) ).

% sum.G_def
tff(fact_3796_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N))) ) ) ) ).

% sum.last_plus
tff(fact_3797_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N))) ) ) ) ).

% prod.last_plus
tff(fact_3798_ex__bij__betw__nat__finite,axiom,
    ! [A: $tType,M5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),M5))
     => ? [H3: fun(nat,A)] : bij_betw(nat,A,H3,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M5)),M5) ) ).

% ex_bij_betw_nat_finite
tff(fact_3799_atLeastLessThanSuc,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,N),set_or7035219750837199246ssThan(nat,M2,N)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThanSuc
tff(fact_3800_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M2: nat,N: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bp(fun(nat,A),fun(nat,A),F3)),set_or7035219750837199246ssThan(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,N)),aa(nat,A,F3,M2)) ) ) ) ).

% sum_Suc_diff'
tff(fact_3801_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,N,M2)) = 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_jb(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M2)),set_or7035219750837199246ssThan(nat,N,M2)) ) ).

% sum.atLeastLessThan_rev
tff(fact_3802_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_jc(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ej(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% sum.nested_swap
tff(fact_3803_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,N,M2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jd(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M2)),set_or7035219750837199246ssThan(nat,N,M2)) ) ).

% prod.atLeastLessThan_rev
tff(fact_3804_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_je(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_em(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% prod.nested_swap
tff(fact_3805_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),K2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jf(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2))) ) ).

% sum.nat_group
tff(fact_3806_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),K2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jg(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2))) ) ).

% prod.nat_group
tff(fact_3807_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_Suc_fact
tff(fact_3808_prod__Suc__fact,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_fact
tff(fact_3809_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N6: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N6),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N6)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_3810_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F3: fun(fun(A,B),C),G3: C] :
          ( ! [X3: fun(A,B)] : aa(fun(A,B),C,F3,X3) = G3
         => ( aa(fun(A,B),C,F3,aTP_Lamp_jh(A,B)) = G3 ) ) ) ).

% fun_cong_unused_0
tff(fact_3811_card__sum__le__nat__sum,axiom,
    ! [S3: set(nat)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_bu(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S3)))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_bu(nat,nat)),S3))) ).

% card_sum_le_nat_sum
tff(fact_3812_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),insert(A,U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_3813_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M2: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G3,N)) ) ) ) ) ).

% sum.head_if
tff(fact_3814_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M2: nat,G3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G3,N)) ) ) ) ) ).

% prod.head_if
tff(fact_3815_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_Suc
tff(fact_3816_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,N,M2)) = 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_bn(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M2)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_3817_atLeastLessThan__nat__numeral,axiom,
    ! [M2: nat,K2: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),pred_numeral(K2)))
       => ( set_or7035219750837199246ssThan(nat,M2,aa(num,nat,numeral_numeral(nat),K2)) = aa(set(nat),set(nat),insert(nat,pred_numeral(K2)),set_or7035219750837199246ssThan(nat,M2,pred_numeral(K2))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),pred_numeral(K2)))
       => ( set_or7035219750837199246ssThan(nat,M2,aa(num,nat,numeral_numeral(nat),K2)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_3818_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat,M2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,N,M2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_dd(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M2)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_3819_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_dk(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% pochhammer_prod
tff(fact_3820_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aa(nat,fun(nat,nat),minus_minus(nat),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_rev
tff(fact_3821_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),S3))
         => ( infini527867602293511546merate(A,S3,aa(nat,nat,suc,N)) = ord_Least(A,aa(nat,fun(A,bool),aTP_Lamp_iw(set(A),fun(nat,fun(A,bool)),S3),N)) ) ) ) ).

% enumerate_Suc''
tff(fact_3822_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [N7: nat] :
                ! [M6: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),M6))
                 => ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,M6,N5)))),E4)) ) ) ) ) ).

% summable_Cauchy
tff(fact_3823_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A),S2: A,K2: nat] :
          ( sums(A,F3,S2)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
           => sums(A,aa(nat,fun(nat,A),aTP_Lamp_ji(fun(nat,A),fun(nat,fun(nat,A)),F3),K2),S2) ) ) ) ).

% sums_group
tff(fact_3824_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_lessThan(nat),N)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_3825_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2),N)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K2))) ) ) ) ).

% fact_split
tff(fact_3826_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S3: set(A),N: nat] : infini527867602293511546merate(A,S3,aa(nat,nat,suc,N)) = infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,ord_Least(A,aTP_Lamp_iz(set(A),fun(A,bool),S3))),bot_bot(set(A)))),N) ) ).

% enumerate_Suc
tff(fact_3827_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jj(nat,fun(nat,fun(nat,A)),K2),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_3828_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,A3),K2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jk(A,fun(nat,fun(nat,A)),A3),K2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ).

% gbinomial_altdef_of_nat
tff(fact_3829_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K2)),semiring_char_0_fact(A,K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jl(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ).

% gbinomial_mult_fact'
tff(fact_3830_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),aa(nat,A,gbinomial(A,A3),K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jl(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ).

% gbinomial_mult_fact
tff(fact_3831_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K2: nat] : aa(nat,A,gbinomial(A,A3),K2) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fs(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)),semiring_char_0_fact(A,K2)) ) ).

% gbinomial_prod_rev
tff(fact_3832_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F3: fun(A,B),I2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_iv(set(A),fun(fun(A,B),fun(A,bool)),I5),F3))))
         => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert(A,I2),bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F3),I5)),aa(A,B,F3,I2)) ) )
            & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert(A,I2),bot_bot(set(A))))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F3),I5) ) ) ) ) ) ).

% sum_diff1'
tff(fact_3833_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_jm(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_3834_xor__int__rec,axiom,
    ! [K2: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K2))),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% xor_int_rec
tff(fact_3835_sum__power2,axiom,
    ! [K2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)),one_one(nat)) ).

% sum_power2
tff(fact_3836_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I3: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A3,I3)),aa(nat,A,A3,J3))) ) )
         => ( ! [I3: nat,J3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J3))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,B2,J3)),aa(nat,A,B2,I3))) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_3837_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A3: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I3: nat,J3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,A3,I3)),aa(nat,nat,A3,J3))) ) )
     => ( ! [I3: nat,J3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J3))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,B2,J3)),aa(nat,nat,B2,I3))) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_jo(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A3),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_3838_xor__int__unfold,axiom,
    ! [K2: int,L: int] :
      ( ( ( K2 = aa(int,int,uminus_uminus(int),one_one(int)) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),L) ) )
      & ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
       => ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),K2) ) )
          & ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( ( ( K2 = zero_zero(int) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),L) = L ) )
              & ( ( K2 != zero_zero(int) )
               => ( ( ( L = zero_zero(int) )
                   => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),L) = K2 ) )
                  & ( ( L != zero_zero(int) )
                   => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),modulo_modulo(int,K2,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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_3839_exp__two__pi__i_H,axiom,
    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),real_Vector_of_real(complex,pi)),aa(num,complex,numeral_numeral(complex),bit0(one2))))) = one_one(complex) ).

% exp_two_pi_i'
tff(fact_3840_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa(set(int),bool,finite_finite2(int),set_or7035219750837199246ssThan(int,L,U))) ).

% finite_atLeastLessThan_int
tff(fact_3841_of__real__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( real_Vector_of_real(A,zero_zero(real)) = zero_zero(A) ) ) ).

% of_real_0
tff(fact_3842_of__real__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real] :
          ( ( real_Vector_of_real(A,X) = zero_zero(A) )
        <=> ( X = zero_zero(real) ) ) ) ).

% of_real_eq_0_iff
tff(fact_3843_of__real__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Y: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),real_Vector_of_real(A,X)),real_Vector_of_real(A,Y)) ) ).

% of_real_mult
tff(fact_3844_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_3845_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_3846_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,L,U)) = nat2(aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ).

% card_atLeastLessThan_int
tff(fact_3847_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_3848_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_3849_sin__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,real_Vector_of_real(A,pi)) = zero_zero(A) ) ) ).

% sin_of_real_pi
tff(fact_3850_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ).

% cos_of_real_pi_half
tff(fact_3851_exp__pi__i_H,axiom,
    exp(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,pi))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i'
tff(fact_3852_exp__pi__i,axiom,
    exp(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,pi)),imaginary_unit)) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i
tff(fact_3853_exp__two__pi__i,axiom,
    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))),real_Vector_of_real(complex,pi))),imaginary_unit)) = one_one(complex) ).

% exp_two_pi_i
tff(fact_3854_complex__exp__exists,axiom,
    ! [Z2: complex] :
    ? [A5: complex,R3: real] : Z2 = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R3)),exp(complex,A5)) ).

% complex_exp_exists
tff(fact_3855_scaleR__conv__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R: real,X: A] : aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),R),X) = aa(A,A,aa(A,fun(A,A),times_times(A),real_Vector_of_real(A,R)),X) ) ).

% scaleR_conv_of_real
tff(fact_3856_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa(set(int),bool,finite_finite2(int),set_or7035219750837199246ssThan(int,zero_zero(int),U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_3857_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,L,aa(int,int,aa(int,fun(int,int),plus_plus(int),U),one_one(int))) = set_or1337092689740270186AtMost(int,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_3858_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(X,Y)),real_Vector_of_real(complex,R)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),X),R),aa(real,real,aa(real,fun(real,real),times_times(real),Y),R)) ).

% Complex_mult_complex_of_real
tff(fact_3859_complex__of__real__mult__Complex,axiom,
    ! [R: real,X: real,Y: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R)),complex2(X,Y)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R),X),aa(real,real,aa(real,fun(real,real),times_times(real),R),Y)) ).

% complex_of_real_mult_Complex
tff(fact_3860_Complex__eq,axiom,
    ! [A3: real,B2: real] : complex2(A3,B2) = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,A3)),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,B2))) ).

% Complex_eq
tff(fact_3861_cis__conv__exp,axiom,
    ! [B2: real] : cis(B2) = exp(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,B2))) ).

% cis_conv_exp
tff(fact_3862_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,zero_zero(int),U)) = nat2(U) ).

% card_atLeastZeroLessThan_int
tff(fact_3863_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M2),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_3864_complex__of__real__i,axiom,
    ! [R: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R)),imaginary_unit) = complex2(zero_zero(real),R) ).

% complex_of_real_i
tff(fact_3865_i__complex__of__real,axiom,
    ! [R: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,R)) = complex2(zero_zero(real),R) ).

% i_complex_of_real
tff(fact_3866_complex__split__polar,axiom,
    ! [Z2: complex] :
    ? [R3: real,A5: real] : Z2 = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R3)),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,cos(real,A5))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,sin(real,A5))))) ).

% complex_split_polar
tff(fact_3867_and__not__numerals_I5_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = 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),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(5)
tff(fact_3868_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_3869_and__not__numerals_I6_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = 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),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(6)
tff(fact_3870_and__not__numerals_I9_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = 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),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(9)
tff(fact_3871_or__not__numerals_I6_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = 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),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% or_not_numerals(6)
tff(fact_3872_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),N))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) )
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N)) ) ) ) ).

% bit_not_iff_eq
tff(fact_3873_cos__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M2: int,X: real] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),M2)),real_Vector_of_real(A,X))) = real_Vector_of_real(A,cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M2)),X))) ) ).

% cos_int_times_real
tff(fact_3874_sin__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M2: int,X: real] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),M2)),real_Vector_of_real(A,X))) = real_Vector_of_real(A,sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M2)),X))) ) ).

% sin_int_times_real
tff(fact_3875_cmod__unit__one,axiom,
    ! [A3: real] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,cos(real,A3))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,sin(real,A3))))) = one_one(real) ).

% cmod_unit_one
tff(fact_3876_cmod__complex__polar,axiom,
    ! [R: real,A3: real] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R)),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,cos(real,A3))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,sin(real,A3)))))) = aa(real,real,abs_abs(real),R) ).

% cmod_complex_polar
tff(fact_3877_or__not__numerals_I5_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = 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),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(5)
tff(fact_3878_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).

% signed_take_bit_def
tff(fact_3879_and__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = 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),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% and_not_numerals(8)
tff(fact_3880_or__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = 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),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(8)
tff(fact_3881_or__not__numerals_I9_J,axiom,
    ! [M2: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = 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),M2)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(9)
tff(fact_3882_not__int__rec,axiom,
    ! [K2: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K2))),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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% not_int_rec
tff(fact_3883_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_3884_Cauchy__iff2,axiom,
    ! [X6: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X6)
    <=> ! [J: nat] :
        ? [M9: nat] :
        ! [M6: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M6))
         => ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),N5))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,X6,M6)),aa(nat,real,X6,N5)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J))))) ) ) ) ).

% Cauchy_iff2
tff(fact_3885_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,vEBT_size_VEBT,X13)),aa(vEBT_VEBT,nat,vEBT_size_VEBT,X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size_gen(1)
tff(fact_3886_VEBT_Osize_I3_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,size_size(vEBT_VEBT),X13)),aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size(3)
tff(fact_3887_size__list__estimation_H,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: nat,F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),aa(A,nat,F3,X)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),size_list(A,F3,Xs))) ) ) ).

% size_list_estimation'
tff(fact_3888_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,nat),G3: fun(A,nat)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X3)),aa(A,nat,G3,X3))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),size_list(A,F3,Xs)),size_list(A,G3,Xs))) ) ).

% size_list_pointwise
tff(fact_3889_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),E3: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
           => ? [M8: nat] :
              ! [M3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M3))
               => ! [N4: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),N4))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,M3)),aa(nat,A,X6,N4)))),E3)) ) ) ) ) ) ).

% CauchyD
tff(fact_3890_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
             => ? [M10: nat] :
                ! [M: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),M))
                 => ! [N3: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),N3))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,M)),aa(nat,A,X6,N3)))),E2)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI
tff(fact_3891_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [M9: nat] :
                ! [M6: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M6))
                 => ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),N5))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,M6)),aa(nat,A,X6,N5)))),E4)) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_3892_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X21: A,X22: list(A)] : size_list(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X21)),size_list(A,X,X22))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_3893_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),nat,finite_card(list(A)),shuffles(A,Xs,Ys)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% card_disjoint_shuffles
tff(fact_3894_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_jp(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_3895_complex__div__cnj,axiom,
    ! [A3: complex,B2: complex] : divide_divide(complex,A3,B2) = divide_divide(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2)),real_Vector_of_real(complex,aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,B2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_div_cnj
tff(fact_3896_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N))
     => ( aa(nat,list(nat),nat_list_decode,aa(nat,nat,suc,N)) = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_jq(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ) ) ).

% list_decode.psimps(2)
tff(fact_3897_complex__cnj__mult,axiom,
    ! [X: complex,Y: complex] : cnj(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),Y)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),cnj(X)),cnj(Y)) ).

% complex_cnj_mult
tff(fact_3898_list__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,list(nat),nat_list_decode,X) = aa(nat,list(nat),nat_list_decode,Y) )
    <=> ( X = Y ) ) ).

% list_decode_eq
tff(fact_3899_times__integer__code_I1_J,axiom,
    ! [K2: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),K2),zero_zero(code_integer)) = zero_zero(code_integer) ).

% times_integer_code(1)
tff(fact_3900_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),zero_zero(code_integer)),L) = zero_zero(code_integer) ).

% times_integer_code(2)
tff(fact_3901_length__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
     => ( aa(list(A),nat,size_size(list(A)),Zs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ) ) ).

% length_shuffles
tff(fact_3902_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_3903_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] : aa(nat,list(nat),nat_list_decode,aa(nat,nat,suc,N)) = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_jq(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ).

% list_decode.simps(2)
tff(fact_3904_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,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% complex_mod_mult_cnj
tff(fact_3905_complex__norm__square,axiom,
    ! [Z2: complex] : real_Vector_of_real(complex,aa(nat,real,aa(real,fun(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_3906_integer__of__int__code,axiom,
    ! [K2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
       => ( code_integer_of_int(K2) = aa(code_integer,code_integer,uminus_uminus(code_integer),code_integer_of_int(aa(int,int,uminus_uminus(int),K2))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K2),zero_zero(int)))
       => ( ( ( K2 = zero_zero(int) )
           => ( code_integer_of_int(K2) = zero_zero(code_integer) ) )
          & ( ( K2 != zero_zero(int) )
           => ( code_integer_of_int(K2) = if(code_integer,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),code_integer_of_int(divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2))))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),code_integer_of_int(divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))))),one_one(code_integer))) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_3907_list__decode_Opelims,axiom,
    ! [X: nat,Y: list(nat)] :
      ( ( aa(nat,list(nat),nat_list_decode,X) = Y )
     => ( accp(nat,nat_list_decode_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = nil(nat) )
             => ~ accp(nat,nat_list_decode_rel,zero_zero(nat)) ) )
         => ~ ! [N3: nat] :
                ( ( X = aa(nat,nat,suc,N3) )
               => ( ( Y = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_jq(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N3)) )
                 => ~ accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N3)) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_3908_list__decode_Oelims,axiom,
    ! [X: nat,Y: list(nat)] :
      ( ( aa(nat,list(nat),nat_list_decode,X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != nil(nat) ) )
       => ~ ! [N3: nat] :
              ( ( X = aa(nat,nat,suc,N3) )
             => ( Y != aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_jq(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N3)) ) ) ) ) ).

% list_decode.elims
tff(fact_3909_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)) = 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_3910_length__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) )
    <=> ( Xs = nil(A) ) ) ).

% length_0_conv
tff(fact_3911_empty__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( nil(A) = replicate(A,N,X) )
    <=> ( N = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_3912_replicate__empty,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( replicate(A,N,X) = nil(A) )
    <=> ( N = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_3913_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A3: A] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_3914_n__lists__Nil,axiom,
    ! [A: $tType,N: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( n_lists(A,N,nil(A)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ) )
      & ( ( N != zero_zero(nat) )
       => ( n_lists(A,N,nil(A)) = nil(list(A)) ) ) ) ).

% n_lists_Nil
tff(fact_3915_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_3916_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_3917_list__encode_Ocases,axiom,
    ! [X: list(nat)] :
      ( ( X != nil(nat) )
     => ~ ! [X3: nat,Xs2: list(nat)] : X != aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),Xs2) ) ).

% list_encode.cases
tff(fact_3918_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_integer_of_int(Xa2)),code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),times_times(int),Xa2),X)) ).

% times_integer.abs_eq
tff(fact_3919_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_3920_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list(A),Ys: list(B),Zs: list(C),Ws: list(D),P2: fun(list(A),fun(list(B),fun(list(C),fun(list(D),bool))))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( ( aa(list(C),nat,size_size(list(C)),Zs) = aa(list(D),nat,size_size(list(D)),Ws) )
         => ( pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P2,nil(A)),nil(B)),nil(C)),nil(D)))
           => ( ! [X3: A,Xs2: list(A),Y3: B,Ys5: list(B),Z: C,Zs2: list(C),W: D,Ws2: list(D)] :
                  ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys5) )
                 => ( ( aa(list(B),nat,size_size(list(B)),Ys5) = aa(list(C),nat,size_size(list(C)),Zs2) )
                   => ( ( aa(list(C),nat,size_size(list(C)),Zs2) = aa(list(D),nat,size_size(list(D)),Ws2) )
                     => ( pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P2,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z),Zs2)),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),W),Ws2))) ) ) ) )
             => pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P2,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_3921_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C),P2: fun(list(A),fun(list(B),fun(list(C),bool)))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P2,nil(A)),nil(B)),nil(C)))
         => ( ! [X3: A,Xs2: list(A),Y3: B,Ys5: list(B),Z: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys5) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys5) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P2,Xs2),Ys5),Zs2))
                   => pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z),Zs2))) ) ) )
           => pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P2,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_3922_list__induct2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P2: fun(list(A),fun(list(B),bool))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,nil(A)),nil(B)))
       => ( ! [X3: A,Xs2: list(A),Y3: B,Ys5: list(B)] :
              ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys5) )
             => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,Xs2),Ys5))
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5))) ) )
         => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P2,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_3923_replicate__0,axiom,
    ! [A: $tType,X: A] : replicate(A,zero_zero(nat),X) = nil(A) ).

% replicate_0
tff(fact_3924_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : size_list(A,X,nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_3925_scaleR__complex_Osimps_I1_J,axiom,
    ! [R: real,X: complex] : re(aa(complex,complex,aa(real,fun(complex,complex),real_V8093663219630862766scaleR(complex),R),X)) = aa(real,real,aa(real,fun(real,real),times_times(real),R),re(X)) ).

% scaleR_complex.simps(1)
tff(fact_3926_list__decode_Osimps_I1_J,axiom,
    aa(nat,list(nat),nat_list_decode,zero_zero(nat)) = nil(nat) ).

% list_decode.simps(1)
tff(fact_3927_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_3928_Re__complex__div__eq__0,axiom,
    ! [A3: complex,B2: complex] :
      ( ( re(divide_divide(complex,A3,B2)) = zero_zero(real) )
    <=> ( re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))) = zero_zero(real) ) ) ).

% Re_complex_div_eq_0
tff(fact_3929_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z2: complex] : real_V7770717601297561774m_norm(complex,Z2) = sqrt(re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z2),cnj(Z2)))) ).

% complex_mod_sqrt_Re_mult_cnj
tff(fact_3930_listrel_Ocases,axiom,
    ! [B: $tType,A: $tType,A12: list(A),A23: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A12),A23)),listrel(A,B,R)))
     => ( ( ( A12 = nil(A) )
         => ( A23 != nil(B) ) )
       => ~ ! [X3: A,Y3: B,Xs2: list(A)] :
              ( ( A12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ! [Ys5: list(B)] :
                  ( ( A23 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5) )
                 => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),R))
                   => ~ pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys5)),listrel(A,B,R))) ) ) ) ) ) ).

% listrel.cases
tff(fact_3931_listrel_Osimps,axiom,
    ! [B: $tType,A: $tType,A12: list(A),A23: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A12),A23)),listrel(A,B,R)))
    <=> ( ( ( A12 = nil(A) )
          & ( A23 = nil(B) ) )
        | ? [X4: A,Y5: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( A23 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4) )
            & pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y5)),R))
            & pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs3),Ys4)),listrel(A,B,R))) ) ) ) ).

% listrel.simps
tff(fact_3932_Re__complex__div__gt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(divide_divide(complex,A3,B2))))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))))) ) ).

% Re_complex_div_gt_0
tff(fact_3933_Re__complex__div__lt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),re(divide_divide(complex,A3,B2))),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2)))),zero_zero(real))) ) ).

% Re_complex_div_lt_0
tff(fact_3934_Re__complex__div__le__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),re(divide_divide(complex,A3,B2))),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2)))),zero_zero(real))) ) ).

% Re_complex_div_le_0
tff(fact_3935_Re__complex__div__ge__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(divide_divide(complex,A3,B2))))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))))) ) ).

% Re_complex_div_ge_0
tff(fact_3936_list__decode_Opsimps_I1_J,axiom,
    ( accp(nat,nat_list_decode_rel,zero_zero(nat))
   => ( aa(nat,list(nat),nat_list_decode,zero_zero(nat)) = nil(nat) ) ) ).

% list_decode.psimps(1)
tff(fact_3937_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A3: real] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),A3)) = re(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A3)),N)) ).

% cos_n_Re_cis_pow_n
tff(fact_3938_complex__add__cnj,axiom,
    ! [Z2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),Z2),cnj(Z2)) = 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_3939_list__encode_Oelims,axiom,
    ! [X: list(nat),Y: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,X) = Y )
     => ( ( ( X = nil(nat) )
         => ( Y != zero_zero(nat) ) )
       => ~ ! [X3: nat,Xs2: list(nat)] :
              ( ( X = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),Xs2) )
             => ( Y != aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),aa(list(nat),nat,nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_3940_Complex__divide,axiom,
    ! [X: complex,Y: complex] : divide_divide(complex,X,Y) = complex2(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,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),divide_divide(real,aa(real,real,aa(real,fun(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,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% Complex_divide
tff(fact_3941_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A)] : aa(nat,A,nth(A,append(A,Xs,append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = X ).

% concat_inth
tff(fact_3942_Re__Reals__divide,axiom,
    ! [R: complex,Z2: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),R),real_Vector_Reals(complex)))
     => ( re(divide_divide(complex,R,Z2)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),re(R)),re(Z2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% Re_Reals_divide
tff(fact_3943_append__eq__append__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Us: list(A),Vs: list(A)] :
      ( ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        | ( aa(list(A),nat,size_size(list(A)),Us) = aa(list(A),nat,size_size(list(A)),Vs) ) )
     => ( ( append(A,Xs,Us) = append(A,Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_3944_in__measures_I1_J,axiom,
    ! [A: $tType,X: A,Y: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measures(A,nil(fun(A,nat))))) ).

% in_measures(1)
tff(fact_3945_list__encode__inverse,axiom,
    ! [X: list(nat)] : aa(nat,list(nat),nat_list_decode,aa(list(nat),nat,nat_list_encode,X)) = X ).

% list_encode_inverse
tff(fact_3946_list__decode__inverse,axiom,
    ! [N: nat] : aa(list(nat),nat,nat_list_encode,aa(nat,list(nat),nat_list_decode,N)) = N ).

% list_decode_inverse
tff(fact_3947_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),append(A,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_append
tff(fact_3948_zip__append,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Us: list(B),Ys: list(A),Vs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Us) )
     => ( zip(A,B,append(A,Xs,Ys),append(B,Us,Vs)) = append(product_prod(A,B),zip(A,B,Xs,Us),zip(A,B,Ys,Vs)) ) ) ).

% zip_append
tff(fact_3949_nth__append__length,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A)] : aa(nat,A,nth(A,append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = X ).

% nth_append_length
tff(fact_3950_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),N: nat] : aa(nat,A,nth(A,append(A,Xs,Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) = aa(nat,A,nth(A,Ys),N) ).

% nth_append_length_plus
tff(fact_3951_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A] : list_update(A,append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys)),aa(list(A),nat,size_size(list(A)),Xs),Y) = append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ).

% list_update_length
tff(fact_3952_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_3953_Im__i__times,axiom,
    ! [Z2: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z2)) = re(Z2) ).

% Im_i_times
tff(fact_3954_real__eq__imaginary__iff,axiom,
    ! [Y: complex,X: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),Y),real_Vector_Reals(complex)))
     => ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),X),real_Vector_Reals(complex)))
       => ( ( X = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Y) )
        <=> ( ( X = zero_zero(complex) )
            & ( Y = zero_zero(complex) ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_3955_imaginary__eq__real__iff,axiom,
    ! [Y: complex,X: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),Y),real_Vector_Reals(complex)))
     => ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),X),real_Vector_Reals(complex)))
       => ( ( aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Y) = X )
        <=> ( ( X = zero_zero(complex) )
            & ( Y = zero_zero(complex) ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_3956_Re__i__times,axiom,
    ! [Z2: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z2)) = aa(real,real,uminus_uminus(real),im(Z2)) ).

% Re_i_times
tff(fact_3957_Reals__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),real_Vector_Reals(A))) ) ).

% Reals_0
tff(fact_3958_list__encode__eq,axiom,
    ! [X: list(nat),Y: list(nat)] :
      ( ( aa(list(nat),nat,nat_list_encode,X) = aa(list(nat),nat,nat_list_encode,Y) )
    <=> ( X = Y ) ) ).

% list_encode_eq
tff(fact_3959_Reals__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),real_Vector_Reals(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),real_Vector_Reals(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),real_Vector_Reals(A))) ) ) ) ).

% Reals_mult
tff(fact_3960_enumerate__append__eq,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : enumerate(A,N,append(A,Xs,Ys)) = append(product_prod(nat,A),enumerate(A,N,Xs),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_3961_nonzero__Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),real_Vector_Reals(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),real_Vector_Reals(A)))
           => ( ( B2 != zero_zero(A) )
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),divide_divide(A,A3,B2)),real_Vector_Reals(A))) ) ) ) ) ).

% nonzero_Reals_divide
tff(fact_3962_nonzero__Reals__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),real_Vector_Reals(A)))
         => ( ( A3 != zero_zero(A) )
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,inverse_inverse(A),A3)),real_Vector_Reals(A))) ) ) ) ).

% nonzero_Reals_inverse
tff(fact_3963_scaleR__complex_Osimps_I2_J,axiom,
    ! [R: real,X: complex] : im(aa(complex,complex,aa(real,fun(complex,complex),real_V8093663219630862766scaleR(complex),R),X)) = aa(real,real,aa(real,fun(real,real),times_times(real),R),im(X)) ).

% scaleR_complex.simps(2)
tff(fact_3964_same__length__different,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != Ys )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => ? [Pre: list(A),X3: A,Xs4: list(A),Y3: A,Ys3: list(A)] :
            ( ( X3 != Y3 )
            & ( Xs = append(A,Pre,append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)),Xs4)) )
            & ( Ys = append(A,Pre,append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A)),Ys3)) ) ) ) ) ).

% same_length_different
tff(fact_3965_list__update__append1,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),Ys: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( list_update(A,append(A,Xs,Ys),I2,X) = append(A,list_update(A,Xs,I2,X),Ys) ) ) ).

% list_update_append1
tff(fact_3966_lex__append__left__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Ys)),append(A,Xs,Zs))),lex(A,R)))
      <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lex(A,R))) ) ) ).

% lex_append_left_iff
tff(fact_3967_lex__append__leftD,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Ys)),append(A,Xs,Zs))),lex(A,R)))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lex(A,R))) ) ) ).

% lex_append_leftD
tff(fact_3968_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R)))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Us)),append(A,Ys,Vs))),lex(A,R))) ) ) ).

% lex_append_rightI
tff(fact_3969_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xs: list(A),R2: set(product_prod(A,A)),Vs: list(A),Ys: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Xs)),lenlex(A,R2)))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Us,Vs)),append(A,Xs,Ys))),lenlex(A,R2))) ) ) ).

% lenlex_append1
tff(fact_3970_list__encode_Osimps_I1_J,axiom,
    aa(list(nat),nat,nat_list_encode,nil(nat)) = zero_zero(nat) ).

% list_encode.simps(1)
tff(fact_3971_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y5: A,Ys4: list(A)] :
          ( ( Xs = append(A,Ys4,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_3972_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),X: A] : aa(list(A),nat,size_size(list(A)),append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_3973_nth__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(nat,A,nth(A,append(A,Xs,Ys)),N) = aa(nat,A,nth(A,Xs),N) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(nat,A,nth(A,append(A,Xs,Ys)),N) = aa(nat,A,nth(A,Ys),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ) ).

% nth_append
tff(fact_3974_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y))) ).

% times_complex.simps(2)
tff(fact_3975_list__update__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A),X: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( list_update(A,append(A,Xs,Ys),N,X) = append(A,list_update(A,Xs,N,X),Ys) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( list_update(A,append(A,Xs,Ys),N,X) = append(A,Xs,list_update(A,Ys,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),X)) ) ) ) ).

% list_update_append
tff(fact_3976_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(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))) ).

% times_complex.simps(1)
tff(fact_3977_Im__complex__div__eq__0,axiom,
    ! [A3: complex,B2: complex] :
      ( ( im(divide_divide(complex,A3,B2)) = zero_zero(real) )
    <=> ( im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))) = zero_zero(real) ) ) ).

% Im_complex_div_eq_0
tff(fact_3978_scaleR__complex_Ocode,axiom,
    ! [R: real,X: complex] : aa(complex,complex,aa(real,fun(complex,complex),real_V8093663219630862766scaleR(complex),R),X) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R),re(X)),aa(real,real,aa(real,fun(real,real),times_times(real),R),im(X))) ).

% scaleR_complex.code
tff(fact_3979_Im__Reals__divide,axiom,
    ! [R: complex,Z2: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),R),real_Vector_Reals(complex)))
     => ( im(divide_divide(complex,R,Z2)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),re(R))),im(Z2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% Im_Reals_divide
tff(fact_3980_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A3: A,Xs: list(B),Ys: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),append(B,Xs,Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(list(B),nat,size_size(list(B)),Xs))),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Ys))) ) ).

% horner_sum_append
tff(fact_3981_Im__complex__div__lt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),im(divide_divide(complex,A3,B2))),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2)))),zero_zero(real))) ) ).

% Im_complex_div_lt_0
tff(fact_3982_Im__complex__div__gt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),im(divide_divide(complex,A3,B2))))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))))) ) ).

% Im_complex_div_gt_0
tff(fact_3983_Im__complex__div__ge__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),im(divide_divide(complex,A3,B2))))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))))) ) ).

% Im_complex_div_ge_0
tff(fact_3984_Im__complex__div__le__0,axiom,
    ! [A3: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),im(divide_divide(complex,A3,B2))),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2)))),zero_zero(real))) ) ).

% Im_complex_div_le_0
tff(fact_3985_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A3: real] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),A3)) = im(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A3)),N)) ).

% sin_n_Im_cis_pow_n
tff(fact_3986_Re__exp,axiom,
    ! [Z2: complex] : re(exp(complex,Z2)) = aa(real,real,aa(real,fun(real,real),times_times(real),exp(real,re(Z2))),cos(real,im(Z2))) ).

% Re_exp
tff(fact_3987_Im__exp,axiom,
    ! [Z2: complex] : im(exp(complex,Z2)) = aa(real,real,aa(real,fun(real,real),times_times(real),exp(real,re(Z2))),sin(real,im(Z2))) ).

% Im_exp
tff(fact_3988_complex__eq,axiom,
    ! [A3: complex] : A3 = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,re(A3))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,im(A3)))) ).

% complex_eq
tff(fact_3989_fun__complex__eq,axiom,
    ! [A: $tType,F3: fun(A,complex),X5: A] : aa(A,complex,F3,X5) = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,re(aa(A,complex,F3,X5)))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,im(aa(A,complex,F3,X5))))) ).

% fun_complex_eq
tff(fact_3990_times__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),Y) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(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(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y)))) ).

% times_complex.code
tff(fact_3991_complex__div__gt__0,axiom,
    ! [A3: complex,B2: complex] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(divide_divide(complex,A3,B2))))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))))) )
      & ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),im(divide_divide(complex,A3,B2))))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B2))))) ) ) ).

% complex_div_gt_0
tff(fact_3992_exp__eq__polar,axiom,
    ! [Z2: complex] : exp(complex,Z2) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,exp(real,re(Z2)))),cis(im(Z2))) ).

% exp_eq_polar
tff(fact_3993_Im__power2,axiom,
    ! [X: complex] : im(aa(nat,complex,aa(complex,fun(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_3994_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list(nat)] : aa(list(nat),nat,nat_list_encode,aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs)) = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),aa(list(nat),nat,nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_3995_series__comparison__complex,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G3: fun(nat,complex),N6: nat,F3: fun(nat,A)] :
          ( summable(complex,G3)
         => ( ! [N3: nat] : pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),aa(nat,complex,G3,N3)),real_Vector_Reals(complex)))
           => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(aa(nat,complex,G3,N3))))
             => ( ! [N3: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N3))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N3))),real_V7770717601297561774m_norm(complex,aa(nat,complex,G3,N3)))) )
               => summable(A,F3) ) ) ) ) ) ).

% series_comparison_complex
tff(fact_3996_complex__diff__cnj,axiom,
    ! [Z2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),Z2),cnj(Z2)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(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_3997_Re__divide,axiom,
    ! [X: complex,Y: complex] : re(divide_divide(complex,X,Y)) = 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,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% Re_divide
tff(fact_3998_complex__mult__cnj,axiom,
    ! [Z2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z2),cnj(Z2)) = real_Vector_of_real(complex,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_mult_cnj
tff(fact_3999_Im__divide,axiom,
    ! [X: complex,Y: complex] : im(divide_divide(complex,X,Y)) = divide_divide(real,aa(real,real,aa(real,fun(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,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% Im_divide
tff(fact_4000_complex__abs__le__norm,axiom,
    ! [Z2: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(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),sqrt(aa(num,real,numeral_numeral(real),bit0(one2)))),real_V7770717601297561774m_norm(complex,Z2)))) ).

% complex_abs_le_norm
tff(fact_4001_csqrt_Ocode,axiom,
    ! [Z2: complex] : csqrt(Z2) = complex2(sqrt(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),if(real,aa(real,bool,aa(real,fun(real,bool),fequal(real),im(Z2)),zero_zero(real)),one_one(real),aa(real,real,sgn_sgn(real),im(Z2)))),sqrt(divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(complex,Z2)),re(Z2)),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% csqrt.code
tff(fact_4002_csqrt_Osimps_I2_J,axiom,
    ! [Z2: complex] : im(csqrt(Z2)) = aa(real,real,aa(real,fun(real,real),times_times(real),if(real,aa(real,bool,aa(real,fun(real,bool),fequal(real),im(Z2)),zero_zero(real)),one_one(real),aa(real,real,sgn_sgn(real),im(Z2)))),sqrt(divide_divide(real,aa(real,real,aa(real,fun(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_4003_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero(real) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),re(X)),zero_zero(real)))
       => ( csqrt(X) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,sqrt(aa(real,real,abs_abs(real),re(X))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_4004_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),im(X)),zero_zero(real)))
        | ( ( im(X) = zero_zero(real) )
          & pp(aa(real,bool,aa(real,fun(real,bool),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_4005_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F3: fun(nat,fun(A,A)),V3: num,N: nat] : aa(nat,A,rec_nat(A,A3,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V3)),N)) = aa(A,A,aa(nat,fun(A,A),F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V3)),N)),aa(nat,A,rec_nat(A,A3,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V3)),N))) ).

% rec_nat_add_eq_if
tff(fact_4006_case__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F3: fun(nat,A),V3: num,N: nat] : case_nat(A,A3,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V3)),N)) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V3)),N)) ).

% case_nat_add_eq_if
tff(fact_4007_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = if(A,aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A3)),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A3)),aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A)))),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A3)) ) ).

% signed_take_bit_code
tff(fact_4008_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_4009_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: A] :
          ( ( aa(A,A,bit_se4730199178511100633sh_bit(A,N),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_4010_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_4011_old_Onat_Osimps_I7_J,axiom,
    ! [T: $tType,F1: T,F22: fun(nat,fun(T,T)),Nat: nat] : aa(nat,T,rec_nat(T,F1,F22),aa(nat,nat,suc,Nat)) = aa(T,T,aa(nat,fun(T,T),F22,Nat),aa(nat,T,rec_nat(T,F1,F22),Nat)) ).

% old.nat.simps(7)
tff(fact_4012_old_Onat_Osimps_I6_J,axiom,
    ! [T: $tType,F1: T,F22: fun(nat,fun(T,T))] : aa(nat,T,rec_nat(T,F1,F22),zero_zero(nat)) = F1 ).

% old.nat.simps(6)
tff(fact_4013_case__nat__numeral,axiom,
    ! [A: $tType,A3: A,F3: fun(nat,A),V3: num] : case_nat(A,A3,F3,aa(num,nat,numeral_numeral(nat),V3)) = aa(nat,A,F3,pred_numeral(V3)) ).

% case_nat_numeral
tff(fact_4014_rec__nat__numeral,axiom,
    ! [A: $tType,A3: A,F3: fun(nat,fun(A,A)),V3: num] : aa(nat,A,rec_nat(A,A3,F3),aa(num,nat,numeral_numeral(nat),V3)) = aa(A,A,aa(nat,fun(A,A),F3,pred_numeral(V3)),aa(nat,A,rec_nat(A,A3,F3),pred_numeral(V3))) ).

% rec_nat_numeral
tff(fact_4015_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,K2: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),K2)) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(num,A,numeral_numeral(A),bit0(K2))) ) ).

% push_bit_Suc_numeral
tff(fact_4016_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K2: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K2))) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K2)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_4017_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% push_bit_of_Suc_0
tff(fact_4018_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),A3) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% push_bit_Suc
tff(fact_4019_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A3)))
        <=> ( ( N != zero_zero(nat) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ) ).

% even_push_bit_iff
tff(fact_4020_nat_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: A,F22: fun(nat,A),Nat: nat] : aa(A,B,H,case_nat(A,F1,F22,Nat)) = case_nat(B,aa(A,B,H,F1),aa(fun(nat,A),fun(nat,B),aTP_Lamp_jr(fun(A,B),fun(fun(nat,A),fun(nat,B)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_4021_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X2: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X2)) = aa(nat,A,F22,X2) ).

% old.nat.simps(5)
tff(fact_4022_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_4023_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> pp(case_nat(bool,fTrue,aTP_Lamp_js(nat,bool),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_4024_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> pp(case_nat(bool,fFalse,aTP_Lamp_jt(nat,bool),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_4025_rec__nat__Suc__imp,axiom,
    ! [A: $tType,F3: fun(nat,A),F1: A,F22: fun(nat,fun(A,A)),N: nat] :
      ( ( F3 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F3,aa(nat,nat,suc,N)) = aa(A,A,aa(nat,fun(A,A),F22,N),aa(nat,A,F3,N)) ) ) ).

% rec_nat_Suc_imp
tff(fact_4026_rec__nat__0__imp,axiom,
    ! [A: $tType,F3: fun(nat,A),F1: A,F22: fun(nat,fun(A,A))] :
      ( ( F3 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F3,zero_zero(nat)) = F1 ) ) ).

% rec_nat_0_imp
tff(fact_4027_bit__push__bit__iff__int,axiom,
    ! [M2: nat,K2: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_se4730199178511100633sh_bit(int,M2),K2)),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ).

% bit_push_bit_iff_int
tff(fact_4028_bit__push__bit__iff__nat,axiom,
    ! [M2: nat,Q2: nat,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M2),Q2)),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_4029_less__eq__nat_Osimps_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N))
    <=> pp(case_nat(bool,fFalse,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_4030_max__Suc2,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M2),aa(nat,nat,suc,N)) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_ju(nat,fun(nat,nat),N),M2) ).

% max_Suc2
tff(fact_4031_max__Suc1,axiom,
    ! [N: nat,M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),M2) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_jv(nat,fun(nat,nat),N),M2) ).

% max_Suc1
tff(fact_4032_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A3)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% push_bit_double
tff(fact_4033_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_4034_diff__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_bu(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)) ).

% diff_Suc
tff(fact_4035_push__bit__nat__def,axiom,
    ! [N: nat,M2: nat] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,N),M2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% push_bit_nat_def
tff(fact_4036_push__bit__int__def,axiom,
    ! [N: nat,K2: int] : aa(int,int,bit_se4730199178511100633sh_bit(int,N),K2) = aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% push_bit_int_def
tff(fact_4037_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% push_bit_eq_mult
tff(fact_4038_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_jw(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% take_bit_sum
tff(fact_4039_old_Orec__nat__def,axiom,
    ! [T: $tType,X5: T,Xa: fun(nat,fun(T,T)),Xb3: nat] : aa(nat,T,rec_nat(T,X5,Xa),Xb3) = the(T,rec_set_nat(T,X5,Xa,Xb3)) ).

% old.rec_nat_def
tff(fact_4040_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,N: nat,X: A,F3: fun(nat,A)] :
      ( ( ( N = zero_zero(nat) )
       => ( case_nat(A,X,F3,N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( case_nat(A,X,F3,N) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).

% Nitpick.case_nat_unfold
tff(fact_4041_nat_Osplit__sels_I2_J,axiom,
    ! [A: $tType,P2: fun(A,bool),F1: A,F22: fun(nat,A),Nat: nat] :
      ( pp(aa(A,bool,P2,case_nat(A,F1,F22,Nat)))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ pp(aa(A,bool,P2,F1)) )
          | ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
            & ~ pp(aa(A,bool,P2,aa(nat,A,F22,pred(Nat)))) ) ) ) ).

% nat.split_sels(2)
tff(fact_4042_nat_Osplit__sels_I1_J,axiom,
    ! [A: $tType,P2: fun(A,bool),F1: A,F22: fun(nat,A),Nat: nat] :
      ( pp(aa(A,bool,P2,case_nat(A,F1,F22,Nat)))
    <=> ( ( ( Nat = zero_zero(nat) )
         => pp(aa(A,bool,P2,F1)) )
        & ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
         => pp(aa(A,bool,P2,aa(nat,A,F22,pred(Nat)))) ) ) ) ).

% nat.split_sels(1)
tff(fact_4043_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_bu(nat,nat),Nat) ).

% pred_def
tff(fact_4044_Suc__0__mod__numeral,axiom,
    ! [K2: num] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),K2)) = aa(product_prod(nat,nat),nat,product_snd(nat,nat),unique8689654367752047608divmod(nat,one2,K2)) ).

% Suc_0_mod_numeral
tff(fact_4045_Suc__0__div__numeral,axiom,
    ! [K2: num] : divide_divide(nat,aa(nat,nat,suc,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),K2)) = aa(product_prod(nat,nat),nat,product_fst(nat,nat),unique8689654367752047608divmod(nat,one2,K2)) ).

% Suc_0_div_numeral
tff(fact_4046_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A3) = A3 ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A3) = aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% drop_bit_rec
tff(fact_4047_nat__of__integer__code,axiom,
    ! [K2: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K2),zero_zero(code_integer)))
       => ( code_nat_of_integer(K2) = zero_zero(nat) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K2),zero_zero(code_integer)))
       => ( code_nat_of_integer(K2) = 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_jx(code_integer,fun(code_integer,nat))),code_divmod_integer(K2,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))) ) ) ) ).

% nat_of_integer_code
tff(fact_4048_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_4049_prod_Ocollapse,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) = Prod ).

% prod.collapse
tff(fact_4050_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,B2: bool] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(bool,A,zero_neq_one_of_bool(A),B2)) = aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat)),B2)) ) ).

% drop_bit_of_bool
tff(fact_4051_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ).

% drop_bit_of_Suc_0
tff(fact_4052_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K2: num] : aa(int,int,bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K2)))) = aa(int,int,bit_se4197421643247451524op_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2))) ).

% drop_bit_Suc_minus_bit0
tff(fact_4053_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K2: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),bit0(K2))) = aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(num,A,numeral_numeral(A),K2)) ) ).

% drop_bit_Suc_bit0
tff(fact_4054_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K2: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K2))) = aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(num,A,numeral_numeral(A),K2)) ) ).

% drop_bit_Suc_bit1
tff(fact_4055_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,N),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% drop_bit_of_1
tff(fact_4056_numeral__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K2: num,L: num] : divide_divide(A,aa(num,A,numeral_numeral(A),K2),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,K2,L)) ) ).

% numeral_div_numeral
tff(fact_4057_numeral__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K2: num,L: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),K2),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,K2,L)) ) ).

% numeral_mod_numeral
tff(fact_4058_fst__divmod__nat,axiom,
    ! [M2: nat,N: nat] : aa(product_prod(nat,nat),nat,product_fst(nat,nat),divmod_nat(M2,N)) = divide_divide(nat,M2,N) ).

% fst_divmod_nat
tff(fact_4059_nat__of__integer__non__positive,axiom,
    ! [K2: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K2),zero_zero(code_integer)))
     => ( code_nat_of_integer(K2) = zero_zero(nat) ) ) ).

% nat_of_integer_non_positive
tff(fact_4060_snd__divmod__nat,axiom,
    ! [M2: nat,N: nat] : aa(product_prod(nat,nat),nat,product_snd(nat,nat),divmod_nat(M2,N)) = modulo_modulo(nat,M2,N) ).

% snd_divmod_nat
tff(fact_4061_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] : aa(int,int,bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K2)))) = aa(int,int,bit_se4197421643247451524op_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K2)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_4062_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).

% one_div_numeral
tff(fact_4063_one__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).

% one_mod_numeral
tff(fact_4064_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType,X5: fun(A,fun(B,C)),Xa: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),X5),Xa) = aa(B,C,aa(A,fun(B,C),X5,aa(product_prod(A,B),A,product_fst(A,B),Xa)),aa(product_prod(A,B),B,product_snd(A,B),Xa)) ).

% case_prod_unfold
tff(fact_4065_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),X5: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),X5) = aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),X5)),aa(product_prod(A,B),B,product_snd(A,B),X5)) ).

% case_prod_beta'
tff(fact_4066_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,A: $tType,D: $tType,F3: fun(A,fun(B,C)),G3: fun(D,A)] : aa(fun(D,A),fun(product_prod(D,B),C),aTP_Lamp_jy(fun(A,fun(B,C)),fun(fun(D,A),fun(product_prod(D,B),C)),F3),G3) = aa(fun(D,fun(B,C)),fun(product_prod(D,B),C),product_case_prod(D,B,C),aa(fun(D,A),fun(D,fun(B,C)),aTP_Lamp_jz(fun(A,fun(B,C)),fun(fun(D,A),fun(D,fun(B,C))),F3),G3)) ).

% split_comp_eq
tff(fact_4067_snd__def,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,aa(fun(A,fun(B,B)),fun(product_prod(A,B),B),product_case_prod(A,B,B),aTP_Lamp_ka(A,fun(B,B))),Prod) ).

% snd_def
tff(fact_4068_The__case__prod,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool))] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2)) = the(product_prod(A,B),aTP_Lamp_kb(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),P2)) ).

% The_case_prod
tff(fact_4069_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_kc(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_4070_prod__eq__iff,axiom,
    ! [B: $tType,A: $tType,S2: product_prod(A,B),T2: product_prod(A,B)] :
      ( ( S2 = T2 )
    <=> ( ( aa(product_prod(A,B),A,product_fst(A,B),S2) = aa(product_prod(A,B),A,product_fst(A,B),T2) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),S2) = aa(product_prod(A,B),B,product_snd(A,B),T2) ) ) ) ).

% prod_eq_iff
tff(fact_4071_prod__eqI,axiom,
    ! [B: $tType,A: $tType,P: product_prod(A,B),Q2: product_prod(A,B)] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),P) = aa(product_prod(A,B),A,product_fst(A,B),Q2) )
     => ( ( aa(product_prod(A,B),B,product_snd(A,B),P) = aa(product_prod(A,B),B,product_snd(A,B),Q2) )
       => ( P = Q2 ) ) ) ).

% prod_eqI
tff(fact_4072_prod_Oexpand,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B),Prod2: product_prod(A,B)] :
      ( ( ( aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,product_fst(A,B),Prod2) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,product_snd(A,B),Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_4073_snd__conv,axiom,
    ! [Aa: $tType,A: $tType,X1: Aa,X2: A] : aa(product_prod(Aa,A),A,product_snd(Aa,A),aa(A,product_prod(Aa,A),aa(Aa,fun(A,product_prod(Aa,A)),product_Pair(Aa,A),X1),X2)) = X2 ).

% snd_conv
tff(fact_4074_snd__eqD,axiom,
    ! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
      ( ( aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = A3 )
     => ( Y = A3 ) ) ).

% snd_eqD
tff(fact_4075_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X2: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2)) = X1 ).

% fst_conv
tff(fact_4076_fst__eqD,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = A3 )
     => ( X = A3 ) ) ).

% fst_eqD
tff(fact_4077_surjective__pairing,axiom,
    ! [B: $tType,A: $tType,T2: product_prod(A,B)] : T2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),T2)),aa(product_prod(A,B),B,product_snd(A,B),T2)) ).

% surjective_pairing
tff(fact_4078_prod_Oexhaust__sel,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).

% prod.exhaust_sel
tff(fact_4079_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool)),X: A,Y: B,A3: product_prod(A,B)] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P2,X),Y))
     => ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) )
       => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(product_prod(A,B),A,product_fst(A,B),A3)),aa(product_prod(A,B),B,product_snd(A,B),A3))) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_4080_Product__Type_OCollect__case__prodD,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B),A6: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A6))))
     => pp(aa(B,bool,aa(A,fun(B,bool),A6,aa(product_prod(A,B),A,product_fst(A,B),X)),aa(product_prod(A,B),B,product_snd(A,B),X))) ) ).

% Product_Type.Collect_case_prodD
tff(fact_4081_case__prod__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,A)),P: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),P) = aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),P)),aa(product_prod(B,C),C,product_snd(B,C),P)) ).

% case_prod_beta
tff(fact_4082_split__beta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),Prod: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),Prod) = aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).

% split_beta
tff(fact_4083_prod_Osplit__sel,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(C,bool),F3: fun(A,fun(B,C)),Prod: product_prod(A,B)] :
      ( pp(aa(C,bool,P2,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),Prod)))
    <=> ( ( Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) )
       => pp(aa(C,bool,P2,aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)))) ) ) ).

% prod.split_sel
tff(fact_4084_prod_Osplit__sel__asm,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(C,bool),F3: fun(A,fun(B,C)),Prod: product_prod(A,B)] :
      ( pp(aa(C,bool,P2,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),Prod)))
    <=> ~ ( ( Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) )
          & ~ pp(aa(C,bool,P2,aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)))) ) ) ).

% prod.split_sel_asm
tff(fact_4085_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = A3 )
        <=> ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A3) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_4086_in__set__zip,axiom,
    ! [A: $tType,B: $tType,P: product_prod(A,B),Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),P),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
    <=> ? [N5: nat] :
          ( ( aa(nat,A,nth(A,Xs),N5) = aa(product_prod(A,B),A,product_fst(A,B),P) )
          & ( aa(nat,B,nth(B,Ys),N5) = aa(product_prod(A,B),B,product_snd(A,B),P) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(B),nat,size_size(list(B)),Ys))) ) ) ).

% in_set_zip
tff(fact_4087_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_4088_fst__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] : aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,M2,N)) = divide_divide(A,aa(num,A,numeral_numeral(A),M2),aa(num,A,numeral_numeral(A),N)) ) ).

% fst_divmod
tff(fact_4089_snd__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M2: num,N: num] : aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,M2,N)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),M2),aa(num,A,numeral_numeral(A),N)) ) ).

% snd_divmod
tff(fact_4090_in__set__enumerate__eq,axiom,
    ! [A: $tType,P: product_prod(nat,A),N: nat,Xs: list(A)] :
      ( pp(aa(set(product_prod(nat,A)),bool,aa(product_prod(nat,A),fun(set(product_prod(nat,A)),bool),member(product_prod(nat,A)),P),aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,N,Xs))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(product_prod(nat,A),nat,product_fst(nat,A),P)))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P)),N)) = aa(product_prod(nat,A),A,product_snd(nat,A),P) ) ) ) ).

% in_set_enumerate_eq
tff(fact_4091_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_4092_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N)),A3) = aa(A,A,bit_se4197421643247451524op_bit(A,N),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% drop_bit_Suc
tff(fact_4093_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),G3: fun(B,nat),P: product_prod(A,B)] : basic_BNF_size_prod(A,B,F3,G3,P) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,aa(product_prod(A,B),A,product_fst(A,B),P))),aa(B,nat,G3,aa(product_prod(A,B),B,product_snd(A,B),P)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_4094_int__of__integer__code,axiom,
    ! [K2: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K2),zero_zero(code_integer)))
       => ( code_int_of_integer(K2) = aa(int,int,uminus_uminus(int),code_int_of_integer(aa(code_integer,code_integer,uminus_uminus(code_integer),K2))) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K2),zero_zero(code_integer)))
       => ( ( ( K2 = zero_zero(code_integer) )
           => ( code_int_of_integer(K2) = zero_zero(int) ) )
          & ( ( K2 != zero_zero(code_integer) )
           => ( code_int_of_integer(K2) = 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_kd(code_integer,fun(code_integer,int))),code_divmod_integer(K2,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_4095_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,bool),P: A,Q: fun(B,bool),Q2: B] :
      ( pp(aa(A,bool,P2,P))
     => ( pp(aa(B,bool,Q,Q2))
       => ( pp(aa(A,bool,P2,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P),Q2))))
          & pp(aa(B,bool,Q,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P),Q2)))) ) ) ) ).

% conjI_realizer
tff(fact_4096_exI__realizer,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool)),Y: A,X: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Y),X))
     => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)))) ) ).

% exI_realizer
tff(fact_4097_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : code_int_of_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),X),Xa2)) = aa(int,int,aa(int,fun(int,int),times_times(int),code_int_of_integer(X)),code_int_of_integer(Xa2)) ).

% times_integer.rep_eq
tff(fact_4098_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y))
     => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Y))))) ) ) ).

% bezw_non_0
tff(fact_4099_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa2) = Y )
     => ( ( ( Xa2 = zero_zero(nat) )
         => ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
        & ( ( Xa2 != zero_zero(nat) )
         => ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Xa2))))) ) ) ) ) ).

% bezw.elims
tff(fact_4100_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero(nat) )
       => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
      & ( ( Y != zero_zero(nat) )
       => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Y))))) ) ) ) ).

% bezw.simps
tff(fact_4101_one__mod__minus__numeral,axiom,
    ! [N: num] : modulo_modulo(int,one_one(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N)))) ).

% one_mod_minus_numeral
tff(fact_4102_minus__one__mod__numeral,axiom,
    ! [N: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(num,int,numeral_numeral(int),N)) = adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N))) ).

% minus_one_mod_numeral
tff(fact_4103_numeral__mod__minus__numeral,axiom,
    ! [M2: num,N: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),M2),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M2,N)))) ).

% numeral_mod_minus_numeral
tff(fact_4104_minus__numeral__mod__numeral,axiom,
    ! [M2: num,N: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)) = adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M2,N))) ).

% minus_numeral_mod_numeral
tff(fact_4105_Divides_Oadjust__mod__def,axiom,
    ! [R: int,L: int] :
      ( ( ( R = zero_zero(int) )
       => ( adjust_mod(L,R) = zero_zero(int) ) )
      & ( ( R != zero_zero(int) )
       => ( adjust_mod(L,R) = aa(int,int,aa(int,fun(int,int),minus_minus(int),L),R) ) ) ) ).

% Divides.adjust_mod_def
tff(fact_4106_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa2) = Y )
     => ( accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2))
       => ~ ( ( ( ( Xa2 = zero_zero(nat) )
               => ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
              & ( ( Xa2 != zero_zero(nat) )
               => ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,modulo_modulo(nat,X,Xa2)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Xa2))))) ) ) )
           => ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2)) ) ) ) ).

% bezw.pelims
tff(fact_4107_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or5935395276787703475ssThan(int,L,U)) = nat2(aa(int,int,aa(int,fun(int,int),minus_minus(int),U),aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).

% card_greaterThanLessThan_int
tff(fact_4108_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X4: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X4),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
           => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X4)) ) ) ) ).

% listrel_iff_zip
tff(fact_4109_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: itself(A)] : semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_ke(nat,bool))) ) ).

% semiring_char_def
tff(fact_4110_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),set_or5935395276787703475ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),U)) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_4111_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa(set(int),bool,finite_finite2(int),set_or5935395276787703475ssThan(int,L,U))) ).

% finite_greaterThanLessThan_int
tff(fact_4112_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K2))
         => ( set_or5935395276787703475ssThan(A,K2,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_4113_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A3,B2) = bot_bot(set(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_4114_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A3,B2) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_4115_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or5935395276787703475ssThan(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Ioo_iff
tff(fact_4116_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or5935395276787703475ssThan(A,A3,B2))) ) ) ).

% infinite_Ioo
tff(fact_4117_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or5935395276787703475ssThan(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_4118_ivl__disj__int__two_I5_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(5)
tff(fact_4119_ivl__disj__int__two_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or5935395276787703475ssThan(A,M2,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(4)
tff(fact_4120_ivl__disj__int__two_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(1)
tff(fact_4121_ivl__disj__int__one_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(1)
tff(fact_4122_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or5935395276787703475ssThan(int,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_4123_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or1337092689740270186AtMost(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_4124_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_4125_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_4126_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A3,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_4127_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),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_4128_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [X4: product_prod(A,A)] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),X4),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Ys))))
           => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),fequal(A)),X4)) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_4129_concat__eq__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ! [X3: product_prod(list(A),list(A))] :
          ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),X3),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys))))
         => pp(aa(product_prod(list(A),list(A)),bool,aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_kf(list(A),fun(list(A),bool))),X3)) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ( concat(A,Xs) = concat(A,Ys) )
        <=> ( Xs = Ys ) ) ) ) ).

% concat_eq_concat_iff
tff(fact_4130_concat__injective,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ( concat(A,Xs) = concat(A,Ys) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ! [X3: product_prod(list(A),list(A))] :
              ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),X3),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys))))
             => pp(aa(product_prod(list(A),list(A)),bool,aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_kf(list(A),fun(list(A),bool))),X3)) )
         => ( Xs = Ys ) ) ) ) ).

% concat_injective
tff(fact_4131_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or5935395276787703475ssThan(A,M2,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_4132_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_4133_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(nat,nat)] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(X),Xa2) = Y )
     => ( accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
               => ( Y = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa2)) ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X))
               => ( Y = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa2),aa(nat,nat,suc,X))) ) ) )
           => ~ accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_4134_bounded__linear__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( ? [K8: real] :
            ! [X3: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K8)))
         => real_V4916620083959148203axioms(A,B,F3) ) ) ).

% bounded_linear_axioms.intro
tff(fact_4135_bounded__linear__axioms__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V4916620083959148203axioms(A,B,F3)
        <=> ? [K6: real] :
            ! [X4: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K6))) ) ) ).

% bounded_linear_axioms_def
tff(fact_4136_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_kg(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_4137_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa(set(nat),bool,finite_finite2(nat),set_or5935395276787703475ssThan(nat,L,U))) ).

% finite_greaterThanLessThan
tff(fact_4138_Suc__funpow,axiom,
    ! [N: nat] : aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),N),suc) = aa(nat,fun(nat,nat),plus_plus(nat),N) ).

% Suc_funpow
tff(fact_4139_funpow__0,axiom,
    ! [A: $tType,F3: fun(A,A),X: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F3),X) = X ).

% funpow_0
tff(fact_4140_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or5935395276787703475ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),aa(nat,nat,suc,L)) ).

% card_greaterThanLessThan
tff(fact_4141_funpow__swap1,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat,X: A] : aa(A,A,F3,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),X)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),aa(A,A,F3,X)) ).

% funpow_swap1
tff(fact_4142_bij__betw__funpow,axiom,
    ! [A: $tType,F3: fun(A,A),S3: set(A),N: nat] :
      ( bij_betw(A,A,F3,S3,S3)
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),S3,S3) ) ).

% bij_betw_funpow
tff(fact_4143_funpow__mult,axiom,
    ! [A: $tType,N: nat,M2: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M2),F3)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),F3) ).

% funpow_mult
tff(fact_4144_funpow__times__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [F3: fun(A,nat),X: A] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(A,nat,F3,X)),aa(A,fun(A,A),times_times(A),X)) = aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(A,nat,F3,X))) ) ).

% funpow_times_power
tff(fact_4145_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,L),U) = set_or5935395276787703475ssThan(nat,L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_4146_numeral__add__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K2: num,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K2)),A3) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K2)),aa(A,fun(A,A),plus_plus(A),one_one(A))),A3) ) ).

% numeral_add_unfold_funpow
tff(fact_4147_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% of_nat_def
tff(fact_4148_numeral__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K2: num] : aa(num,A,numeral_numeral(A),K2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K2)),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% numeral_unfold_funpow
tff(fact_4149_relpowp__bot,axiom,
    ! [A: $tType,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),bot_bot(fun(A,fun(A,bool)))) = bot_bot(fun(A,fun(A,bool))) ) ) ).

% relpowp_bot
tff(fact_4150_relpowp__fun__conv,axiom,
    ! [A: $tType,N: nat,P2: fun(A,fun(A,bool)),X: A,Y: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P2),X),Y))
    <=> ? [F5: fun(nat,A)] :
          ( ( aa(nat,A,F5,zero_zero(nat)) = X )
          & ( aa(nat,A,F5,N) = Y )
          & ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),P2,aa(nat,A,F5,I)),aa(nat,A,F5,aa(nat,nat,suc,I)))) ) ) ) ).

% relpowp_fun_conv
tff(fact_4151_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_4152_sub__num__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [L: num] : neg_numeral_sub(A,one2,bit0(L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bitM(L))) ) ).

% sub_num_simps(2)
tff(fact_4153_sub__num__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_sub(A,one2,one2) = zero_zero(A) ) ) ).

% sub_num_simps(1)
tff(fact_4154_diff__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,M2,N) ) ).

% diff_numeral_simps(1)
tff(fact_4155_sub__num__simps_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num,L: num] : neg_numeral_sub(A,bit0(K2),bit0(L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K2,L)) ) ).

% sub_num_simps(6)
tff(fact_4156_sub__num__simps_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num,L: num] : neg_numeral_sub(A,aa(num,num,bit1,K2),aa(num,num,bit1,L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K2,L)) ) ).

% sub_num_simps(9)
tff(fact_4157_add__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,M2) ) ).

% add_neg_numeral_simps(2)
tff(fact_4158_add__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,M2,N) ) ).

% add_neg_numeral_simps(1)
tff(fact_4159_diff__numeral__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,M2) ) ).

% diff_numeral_simps(4)
tff(fact_4160_sub__num__simps_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num,L: num] : neg_numeral_sub(A,bit0(K2),aa(num,num,bit1,L)) = neg_numeral_dbl_dec(A,neg_numeral_sub(A,K2,L)) ) ).

% sub_num_simps(7)
tff(fact_4161_sub__num__simps_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num,L: num] : neg_numeral_sub(A,aa(num,num,bit1,K2),bit0(L)) = neg_numeral_dbl_inc(A,neg_numeral_sub(A,K2,L)) ) ).

% sub_num_simps(8)
tff(fact_4162_diff__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M2)),one_one(A)) = neg_numeral_sub(A,M2,one2) ) ).

% diff_numeral_special(2)
tff(fact_4163_diff__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,one2,N) ) ).

% diff_numeral_special(1)
tff(fact_4164_sub__num__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_sub(A,aa(num,num,bit1,K2),one2) = aa(num,A,numeral_numeral(A),bit0(K2)) ) ).

% sub_num_simps(5)
tff(fact_4165_sub__num__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_sub(A,bit0(K2),one2) = aa(num,A,numeral_numeral(A),bitM(K2)) ) ).

% sub_num_simps(4)
tff(fact_4166_add__neg__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,one2) ) ).

% add_neg_numeral_special(4)
tff(fact_4167_add__neg__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M2)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,M2,one2) ) ).

% add_neg_numeral_special(3)
tff(fact_4168_add__neg__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),one_one(A)) = neg_numeral_sub(A,one2,M2) ) ).

% add_neg_numeral_special(2)
tff(fact_4169_add__neg__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))) = neg_numeral_sub(A,one2,M2) ) ).

% add_neg_numeral_special(1)
tff(fact_4170_diff__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2))),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,one2,M2) ) ).

% diff_numeral_special(8)
tff(fact_4171_diff__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,one2) ) ).

% diff_numeral_special(7)
tff(fact_4172_minus__sub__one__diff__one,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M2: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),neg_numeral_sub(A,M2,one2))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M2)) ) ).

% minus_sub_one_diff_one
tff(fact_4173_sub__num__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [L: num] : neg_numeral_sub(A,one2,aa(num,num,bit1,L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(L))) ) ).

% sub_num_simps(3)
tff(fact_4174_neg__numeral__class_Osub__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num,L: num] : neg_numeral_sub(A,K2,L) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),K2)),aa(num,A,numeral_numeral(A),L)) ) ).

% neg_numeral_class.sub_def
tff(fact_4175_relpowp__Suc__E,axiom,
    ! [A: $tType,N: nat,P2: fun(A,fun(A,bool)),X: A,Z2: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P2),X),Z2))
     => ~ ! [Y3: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P2),X),Y3))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),P2,Y3),Z2)) ) ) ).

% relpowp_Suc_E
tff(fact_4176_relpowp__Suc__I,axiom,
    ! [A: $tType,N: nat,P2: fun(A,fun(A,bool)),X: A,Y: A,Z2: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P2),X),Y))
     => ( pp(aa(A,bool,aa(A,fun(A,bool),P2,Y),Z2))
       => pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P2),X),Z2)) ) ) ).

% relpowp_Suc_I
tff(fact_4177_relpowp__Suc__D2,axiom,
    ! [A: $tType,N: nat,P2: fun(A,fun(A,bool)),X: A,Z2: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P2),X),Z2))
     => ? [Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),P2,X),Y3))
          & pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P2),Y3),Z2)) ) ) ).

% relpowp_Suc_D2
tff(fact_4178_relpowp__Suc__E2,axiom,
    ! [A: $tType,N: nat,P2: fun(A,fun(A,bool)),X: A,Z2: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P2),X),Z2))
     => ~ ! [Y3: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),P2,X),Y3))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P2),Y3),Z2)) ) ) ).

% relpowp_Suc_E2
tff(fact_4179_relpowp__Suc__I2,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),X: A,Y: A,N: nat,Z2: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),P2,X),Y))
     => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P2),Y),Z2))
       => pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P2),X),Z2)) ) ) ).

% relpowp_Suc_I2
tff(fact_4180_relpowp_Osimps_I1_J,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),R2) = fequal(A) ).

% relpowp.simps(1)
tff(fact_4181_relpowp__0__E,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),X: A,Y: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),P2),X),Y))
     => ( X = Y ) ) ).

% relpowp_0_E
tff(fact_4182_relpowp__0__I,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),X: A] : pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),P2),X),X)) ).

% relpowp_0_I
tff(fact_4183_sub__non__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),neg_numeral_sub(A,N,M2)),zero_zero(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M2)) ) ) ).

% sub_non_positive
tff(fact_4184_sub__non__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),neg_numeral_sub(A,N,M2)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M2),N)) ) ) ).

% sub_non_negative
tff(fact_4185_sub__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),neg_numeral_sub(A,N,M2)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N)) ) ) ).

% sub_positive
tff(fact_4186_sub__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M2: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),neg_numeral_sub(A,N,M2)),zero_zero(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M2)) ) ) ).

% sub_negative
tff(fact_4187_sub__inc__One__eq,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : neg_numeral_sub(A,inc(N),one2) = aa(num,A,numeral_numeral(A),N) ) ).

% sub_inc_One_eq
tff(fact_4188_sub__BitM__One__eq,axiom,
    ! [N: num] : neg_numeral_sub(int,bitM(N),one2) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),neg_numeral_sub(int,N,one2)) ).

% sub_BitM_One_eq
tff(fact_4189_relpowp__E2,axiom,
    ! [A: $tType,N: nat,P2: fun(A,fun(A,bool)),X: A,Z2: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P2),X),Z2))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z2 ) )
       => ~ ! [Y3: A,M: nat] :
              ( ( N = aa(nat,nat,suc,M) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),P2,X),Y3))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),M),P2),Y3),Z2)) ) ) ) ) ).

% relpowp_E2
tff(fact_4190_relpowp__E,axiom,
    ! [A: $tType,N: nat,P2: fun(A,fun(A,bool)),X: A,Z2: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P2),X),Z2))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z2 ) )
       => ~ ! [Y3: A,M: nat] :
              ( ( N = aa(nat,nat,suc,M) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),M),P2),X),Y3))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),P2,Y3),Z2)) ) ) ) ) ).

% relpowp_E
tff(fact_4191_bij__betw__nth__root__unity,axiom,
    ! [C3: complex,N: nat] :
      ( ( C3 != zero_zero(complex) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => bij_betw(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,aa(real,real,root(N),real_V7770717601297561774m_norm(complex,C3)))),cis(divide_divide(real,arg(C3),aa(nat,real,semiring_1_of_nat(real),N))))),aa(fun(complex,bool),set(complex),collect(complex),aTP_Lamp_an(nat,fun(complex,bool),N)),aa(fun(complex,bool),set(complex),collect(complex),aa(nat,fun(complex,bool),aTP_Lamp_ao(complex,fun(nat,fun(complex,bool)),C3),N))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_4192_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_kh(A,fun(A,bool)),aTP_Lamp_ki(A,fun(A,bool))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_4193_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_ac(nat,fun(nat,bool)),aTP_Lamp_bt(nat,fun(nat,bool))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_4194_bit_Oabstract__boolean__algebra__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => boolea2506097494486148201lgebra(A,bit_se5824344872417868541ns_and(A),bit_se1065995026697491101ons_or(A),bit_ri4277139882892585799ns_not(A),zero_zero(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% bit.abstract_boolean_algebra_axioms
tff(fact_4195_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_4196_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = aa(real,real,root(N),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_4197_root__0,axiom,
    ! [X: real] : aa(real,real,root(zero_zero(nat)),X) = zero_zero(real) ).

% root_0
tff(fact_4198_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_4199_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ).

% real_root_less_iff
tff(fact_4200_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ) ).

% real_root_le_iff
tff(fact_4201_real__root__one,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_4202_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = one_one(real) )
      <=> ( X = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_4203_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y)) ) ) ).

% real_root_gt_0_iff
tff(fact_4204_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ) ).

% real_root_lt_0_iff
tff(fact_4205_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y)) ) ) ).

% real_root_ge_0_iff
tff(fact_4206_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ) ).

% real_root_le_0_iff
tff(fact_4207_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),Y)) ) ) ).

% real_root_gt_1_iff
tff(fact_4208_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),one_one(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ) ).

% real_root_lt_1_iff
tff(fact_4209_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y)) ) ) ).

% real_root_ge_1_iff
tff(fact_4210_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),one_one(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) ) ) ).

% real_root_le_1_iff
tff(fact_4211_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_4212_real__root__mult__exp,axiom,
    ! [M2: nat,N: nat,X: real] : aa(real,real,root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),X) = aa(real,real,root(M2),aa(real,real,root(N),X)) ).

% real_root_mult_exp
tff(fact_4213_real__root__mult,axiom,
    ! [N: nat,X: real,Y: real] : aa(real,real,root(N),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(N),X)),aa(real,real,root(N),Y)) ).

% real_root_mult
tff(fact_4214_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y))) ) ) ).

% real_root_less_mono
tff(fact_4215_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y))) ) ) ).

% real_root_le_mono
tff(fact_4216_real__root__power,axiom,
    ! [N: nat,X: real,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),K2)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),K2) ) ) ).

% real_root_power
tff(fact_4217_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(real,real,abs_abs(real),X)) = aa(real,real,abs_abs(real),aa(real,real,root(N),X)) ) ) ).

% real_root_abs
tff(fact_4218_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,sgn_sgn(real),aa(real,real,root(N),X)) = aa(real,real,sgn_sgn(real),X) ) ) ).

% sgn_root
tff(fact_4219_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,root(N),X))) ) ) ).

% real_root_gt_zero
tff(fact_4220_real__root__strict__decreasing,axiom,
    ! [N: nat,N6: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N6))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N6),X)),aa(real,real,root(N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_4221_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,abs_abs(real),aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),N))) = aa(real,real,abs_abs(real),Y) ) ) ).

% root_abs_power
tff(fact_4222_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,root(N),X))) ) ) ).

% real_root_pos_pos
tff(fact_4223_real__root__strict__increasing,axiom,
    ! [N: nat,N6: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N6))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N6),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_4224_real__root__decreasing,axiom,
    ! [N: nat,N6: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N6))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N6),X)),aa(real,real,root(N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_4225_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_4226_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),N) = X )
         => ( aa(real,real,root(N),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_4227_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_4228_real__root__increasing,axiom,
    ! [N: nat,N6: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N6))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N6),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_4229_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),aa(real,real,root(N),X))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),aa(real,real,root(N),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_4230_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_4231_ln__root,axiom,
    ! [N: nat,B2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
       => ( ln_ln(real,aa(real,real,root(N),B2)) = divide_divide(real,ln_ln(real,B2),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% ln_root
tff(fact_4232_log__root,axiom,
    ! [N: nat,A3: real,B2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
       => ( aa(real,real,log2(B2),aa(real,real,root(N),A3)) = divide_divide(real,aa(real,real,log2(B2),A3),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% log_root
tff(fact_4233_log__base__root,axiom,
    ! [N: nat,B2: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
       => ( aa(real,real,log2(aa(real,real,root(N),B2)),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log2(B2),X)) ) ) ) ).

% log_base_root
tff(fact_4234_split__root,axiom,
    ! [P2: fun(real,bool),N: nat,X: real] :
      ( pp(aa(real,bool,P2,aa(real,real,root(N),X)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(real,bool,P2,zero_zero(real))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ! [Y5: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y5)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y5)),N)) = X )
             => pp(aa(real,bool,P2,Y5)) ) ) ) ) ).

% split_root
tff(fact_4235_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( aa(real,real,root(N),X) = powr(real,X,divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% root_powr_inverse
tff(fact_4236_eq__snd__iff,axiom,
    ! [B: $tType,A: $tType,B2: A,P: product_prod(B,A)] :
      ( ( B2 = aa(product_prod(B,A),A,product_snd(B,A),P) )
    <=> ? [A7: B] : P = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A7),B2) ) ).

% eq_snd_iff
tff(fact_4237_sndI,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),Y: A,Z2: B] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z2) )
     => ( aa(product_prod(A,B),B,product_snd(A,B),X) = Z2 ) ) ).

% sndI
tff(fact_4238_eq__fst__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,P: product_prod(A,B)] :
      ( ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P) )
    <=> ? [B5: B] : P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B5) ) ).

% eq_fst_iff
tff(fact_4239_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_4240_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z2: A] : powr(A,zero_zero(A),Z2) = zero_zero(A) ) ).

% powr_0
tff(fact_4241_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( powr(A,X,zero_zero(A)) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( powr(A,X,zero_zero(A)) = one_one(A) ) ) ) ) ).

% powr_zero_eq_one
tff(fact_4242_powr__powr,axiom,
    ! [X: real,A3: real,B2: real] : powr(real,powr(real,X,A3),B2) = powr(real,X,aa(real,real,aa(real,fun(real,real),times_times(real),A3),B2)) ).

% powr_powr
tff(fact_4243_powr__mult,axiom,
    ! [X: real,Y: real,A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y),A3) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,X,A3)),powr(real,Y,A3)) ) ) ) ).

% powr_mult
tff(fact_4244_divide__powr__uminus,axiom,
    ! [A3: real,B2: real,C3: real] : divide_divide(real,A3,powr(real,B2,C3)) = aa(real,real,aa(real,fun(real,real),times_times(real),A3),powr(real,B2,aa(real,real,uminus_uminus(real),C3))) ).

% divide_powr_uminus
tff(fact_4245_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero(real) )
     => ( ln_ln(real,powr(real,X,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),ln_ln(real,X)) ) ) ).

% ln_powr
tff(fact_4246_log__powr,axiom,
    ! [X: real,B2: real,Y: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,log2(B2),powr(real,X,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,log2(B2),X)) ) ) ).

% log_powr
tff(fact_4247_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,A3: A,B2: A] : powr(A,X,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),powr(A,X,A3)),powr(A,X,B2)) ) ).

% powr_add
tff(fact_4248_prod_Osize__neq,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B)] : aa(product_prod(A,B),nat,size_size(product_prod(A,B)),X) != zero_zero(nat) ).

% prod.size_neq
tff(fact_4249_sum_Osize__neq,axiom,
    ! [A: $tType,B: $tType,X: sum_sum(A,B)] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),X) != zero_zero(nat) ).

% sum.size_neq
tff(fact_4250_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),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_4251_ln__powr__bound2,axiom,
    ! [X: real,A3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A3))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,ln_ln(real,X),A3)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A3,A3)),X))) ) ) ).

% ln_powr_bound2
tff(fact_4252_log__add__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log2(B2),X)),Y) = aa(real,real,log2(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_4253_add__log__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),aa(real,real,log2(B2),X)) = aa(real,real,log2(B2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_4254_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A,A3: A] :
          ( ( ( X = zero_zero(A) )
           => ( powr(A,X,A3) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( powr(A,X,A3) = exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),ln_ln(A,X))) ) ) ) ) ).

% powr_def
tff(fact_4255_log__minus__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log2(B2),X)),Y) = aa(real,real,log2(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_4256_fstI,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B),Y: A,Z2: B] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z2) )
     => ( aa(product_prod(A,B),A,product_fst(A,B),X) = Y ) ) ).

% fstI
tff(fact_4257_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% times_int.abs_eq
tff(fact_4258_eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y))) ) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_4259_eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = one_one(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2))) ) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_4260_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,suc,N)) = inc(num_of_nat(N)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,suc,N)) = one2 ) ) ) ).

% num_of_nat.simps(2)
tff(fact_4261_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : num_of_nat(aa(num,nat,numeral_numeral(nat),Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_4262_iszero__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W2: num] :
          ( ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),W2)) ) ) ).

% iszero_neg_numeral
tff(fact_4263_eq__Abs__Integ,axiom,
    ! [Z2: int] :
      ~ ! [X3: nat,Y3: nat] : Z2 != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),Y3)) ).

% eq_Abs_Integ
tff(fact_4264_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_4265_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z2: A] :
          ( ring_1_iszero(A,Z2)
        <=> ( Z2 = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_4266_not__iszero__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [W2: num] : ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),W2)) ) ).

% not_iszero_numeral
tff(fact_4267_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_4268_eq__iff__iszero__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ring_1_iszero(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) ) ) ).

% eq_iff_iszero_diff
tff(fact_4269_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_4270_eq__numeral__iff__iszero_I10_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(A) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).

% eq_numeral_iff_iszero(10)
tff(fact_4271_eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),X)) ) ) ).

% eq_numeral_iff_iszero(9)
tff(fact_4272_not__iszero__Numeral1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),one2)) ) ).

% not_iszero_Numeral1
tff(fact_4273_not__iszero__neg__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_iszero_neg_1
tff(fact_4274_eq__numeral__iff__iszero_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,X,Y)) ) ) ).

% eq_numeral_iff_iszero(1)
tff(fact_4275_zero__int__def,axiom,
    zero_zero(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))) ).

% zero_int_def
tff(fact_4276_int__def,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),N) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N),zero_zero(nat))) ).

% int_def
tff(fact_4277_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(num,nat,numeral_numeral(nat),num_of_nat(N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_4278_eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_4279_eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),X)) ) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_4280_not__iszero__neg__Numeral1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) ) ).

% not_iszero_neg_Numeral1
tff(fact_4281_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),one_one(nat)))
     => ( num_of_nat(N) = one2 ) ) ).

% num_of_nat_One
tff(fact_4282_eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y))) ) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_4283_eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y))) ) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_4284_eq__numeral__iff__iszero_I4_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Y,X)) ) ) ).

% eq_numeral_iff_iszero(4)
tff(fact_4285_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod(nat,nat)] : aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_kl(nat,fun(nat,product_prod(nat,nat)))),X)) ).

% uminus_int.abs_eq
tff(fact_4286_one__int__def,axiom,
    one_one(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))) ).

% one_int_def
tff(fact_4287_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(num,A,numeral_numeral(A),num_of_nat(N)) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(num,A,numeral_numeral(A),num_of_nat(N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ) ) ) ).

% numeral_num_of_nat_unfold
tff(fact_4288_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)) = bit0(num_of_nat(N)) ) ) ).

% num_of_nat_double
tff(fact_4289_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(M2)),num_of_nat(N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_4290_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_kn(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa2),X)) ) ).

% less_eq_int.abs_eq
tff(fact_4291_eq__numeral__iff__iszero_I5_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = one_one(A) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,X,one2)) ) ) ).

% eq_numeral_iff_iszero(5)
tff(fact_4292_eq__numeral__iff__iszero_I6_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,one2,Y)) ) ) ).

% eq_numeral_iff_iszero(6)
tff(fact_4293_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% plus_int.abs_eq
tff(fact_4294_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X)) ).

% minus_int.abs_eq
tff(fact_4295_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)))
    <=> ? [Y5: A,N5: nat] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N5)),Y5)),R))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( Ys = list_update(A,Xs,N5,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_4296_prod_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),P: fun(B,A),I2: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ap(set(B),fun(fun(B,A),fun(B,bool)),I5),P))))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P),aa(set(B),set(B),insert(B,I2),I5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P),I5) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P),aa(set(B),set(B),insert(B,I2),I5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,P,I2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P),I5)) ) ) ) ) ) ).

% prod.insert'
tff(fact_4297_sorted__list__of__set__atMost__Suc,axiom,
    ! [K2: nat] : linord4507533701916653071of_set(nat,aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K2))) = append(nat,linord4507533701916653071of_set(nat,aa(nat,set(nat),set_ord_atMost(nat),K2)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,K2)),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_4298_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K2: nat] : linord4507533701916653071of_set(nat,aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K2))) = append(nat,linord4507533701916653071of_set(nat,aa(nat,set(nat),set_ord_lessThan(nat),K2)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),K2),nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_4299_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] : aa(list(A),nat,size_size(list(A)),linord4507533701916653071of_set(A,A6)) = aa(set(A),nat,finite_card(A),A6) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_4300_Cons__listrel1__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),listrel1(A,R)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R))) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_4301_listrel1__eq__len,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).

% listrel1_eq_len
tff(fact_4302_prod_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_cy(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G3),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),I5)) ) ) ) ).

% prod.distrib_triv'
tff(fact_4303_listrel1I1,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A)),Xs: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))),listrel1(A,R))) ) ).

% listrel1I1
tff(fact_4304_Cons__listrel1E1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys)),listrel1(A,R)))
     => ( ! [Y3: A] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Xs) )
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R)) )
       => ~ ! [Zs2: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2) )
             => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2)),listrel1(A,R))) ) ) ) ).

% Cons_listrel1E1
tff(fact_4305_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),listrel1(A,R)))
     => ( ! [X3: A] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys) )
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y)),R)) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2) )
             => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Zs2),Ys)),listrel1(A,R))) ) ) ) ).

% Cons_listrel1E2
tff(fact_4306_prod_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),G3: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ap(set(B),fun(fun(B,A),fun(B,bool)),I5),G3))))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_ap(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_cy(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G3),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_4307_listrel1E,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)))
     => ~ ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R))
           => ! [Us2: list(A),Vs2: list(A)] :
                ( ( Xs = append(A,Us2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Vs2)) )
               => ( Ys != append(A,Us2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Vs2)) ) ) ) ) ).

% listrel1E
tff(fact_4308_listrel1I,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A)),Xs: list(A),Us: list(A),Vs: list(A),Ys: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
     => ( ( Xs = append(A,Us,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs)) )
       => ( ( Ys = append(A,Us,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Vs)) )
         => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R))) ) ) ) ).

% listrel1I
tff(fact_4309_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),J2))
     => ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I2,J2)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I2)),linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I2),J2))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_4310_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))),append(A,Ys,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),listrel1(A,R)))
    <=> ( ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)))
          & ( X = Y ) )
        | ( ( Xs = Ys )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R)) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_4311_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),aa(nat,nat,suc,I2))))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I2,J2))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_4312_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa2))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_kn(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa2))) ) ).

% less_eq_int.rep_eq
tff(fact_4313_listrel__def,axiom,
    ! [B: $tType,A: $tType,X5: set(product_prod(A,B))] : listrel(A,B,X5) = aa(fun(product_prod(list(A),list(B)),bool),set(product_prod(list(A),list(B))),collect(product_prod(list(A),list(B))),aa(fun(list(A),fun(list(B),bool)),fun(product_prod(list(A),list(B)),bool),product_case_prod(list(A),list(B),bool),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),X5)))) ).

% listrel_def
tff(fact_4314_nths__Cons,axiom,
    ! [A: $tType,X: A,L: list(A),A6: set(nat)] : nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L),A6) = append(A,if(list(A),aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),nil(A)),nths(A,L,aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_ks(set(nat),fun(nat,bool),A6)))) ).

% nths_Cons
tff(fact_4315_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C3: nat,Y: nat,X: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C3),Y))
       => ( image2(nat,nat,aTP_Lamp_kt(nat,fun(nat,nat),C3),set_or7035219750837199246ssThan(nat,X,Y)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),C3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C3)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C3),Y))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
           => ( image2(nat,nat,aTP_Lamp_kt(nat,fun(nat,nat),C3),set_or7035219750837199246ssThan(nat,X,Y)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
           => ( image2(nat,nat,aTP_Lamp_kt(nat,fun(nat,nat),C3),set_or7035219750837199246ssThan(nat,X,Y)) = bot_bot(set(nat)) ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_4316_bij__betw__Suc,axiom,
    ! [M5: set(nat),N6: set(nat)] :
      ( bij_betw(nat,nat,suc,M5,N6)
    <=> ( image2(nat,nat,suc,M5) = N6 ) ) ).

% bij_betw_Suc
tff(fact_4317_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A,J2: A] : image2(A,A,aa(A,fun(A,A),plus_plus(A),K2),set_or1337092689740270186AtMost(A,I2,J2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K2)) ) ).

% image_add_atLeastAtMost
tff(fact_4318_image__diff__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A3: A,B2: A] : image2(A,A,aa(A,fun(A,A),minus_minus(A),D3),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),D3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),D3),A3)) ) ).

% image_diff_atLeastAtMost
tff(fact_4319_image__uminus__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A] : image2(A,A,uminus_uminus(A),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_atLeastAtMost
tff(fact_4320_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A,J2: A] : image2(A,A,aa(A,fun(A,A),plus_plus(A),K2),set_or7035219750837199246ssThan(A,I2,J2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K2)) ) ).

% image_add_atLeastLessThan
tff(fact_4321_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C3: A,A3: A] : image2(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,set(A),set_ord_atMost(A),A3)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)) ) ).

% image_add_atMost
tff(fact_4322_image__uminus__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A] : image2(A,A,uminus_uminus(A),set_or5935395276787703475ssThan(A,X,Y)) = set_or5935395276787703475ssThan(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_greaterThanLessThan
tff(fact_4323_image__Suc__atLeastAtMost,axiom,
    ! [I2: nat,J2: nat] : image2(nat,nat,suc,set_or1337092689740270186AtMost(nat,I2,J2)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I2),aa(nat,nat,suc,J2)) ).

% image_Suc_atLeastAtMost
tff(fact_4324_image__Suc__atLeastLessThan,axiom,
    ! [I2: nat,J2: nat] : image2(nat,nat,suc,set_or7035219750837199246ssThan(nat,I2,J2)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I2),aa(nat,nat,suc,J2)) ).

% image_Suc_atLeastLessThan
tff(fact_4325_bij__betw__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N6: set(nat),A6: set(A)] :
          ( bij_betw(nat,A,semiring_1_of_nat(A),N6,A6)
        <=> ( image2(nat,A,semiring_1_of_nat(A),N6) = A6 ) ) ) ).

% bij_betw_of_nat
tff(fact_4326_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A,J2: A] : image2(A,A,aTP_Lamp_ku(A,fun(A,A),K2),set_or1337092689740270186AtMost(A,I2,J2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K2)) ) ).

% image_add_atLeastAtMost'
tff(fact_4327_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A3: A,B2: A] : image2(A,A,aTP_Lamp_kv(A,fun(A,A),D3),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),D3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_4328_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A,J2: A] : image2(A,A,aTP_Lamp_ku(A,fun(A,A),K2),set_or7035219750837199246ssThan(A,I2,J2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J2),K2)) ) ).

% image_add_atLeastLessThan'
tff(fact_4329_nths__singleton,axiom,
    ! [A: $tType,A6: set(nat),X: A] :
      ( ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A6))
       => ( nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),A6) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) )
      & ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A6))
       => ( nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),A6) = nil(A) ) ) ) ).

% nths_singleton
tff(fact_4330_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D3))
         => ( image2(A,A,aa(A,fun(A,A),times_times(A),D3),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),D3),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_4331_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D3))
         => ( image2(A,A,aTP_Lamp_kw(A,fun(A,A),D3),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,divide_divide(A,A3,D3),divide_divide(A,B2,D3)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_4332_zero__notin__Suc__image,axiom,
    ! [A6: set(nat)] : ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),image2(nat,nat,suc,A6))) ).

% zero_notin_Suc_image
tff(fact_4333_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),I3),I5)) )
     => ( nths(A,Xs,I5) = Xs ) ) ).

% nths_all
tff(fact_4334_card__image__le,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),image2(A,B,F3,A6))),aa(set(A),nat,finite_card(A),A6))) ) ).

% card_image_le
tff(fact_4335_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),image2(A,B,F3,A6)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B6)),aa(set(A),nat,finite_card(A),A6))) ) ) ).

% surj_card_le
tff(fact_4336_image__Suc__lessThan,axiom,
    ! [N: nat] : image2(nat,nat,suc,aa(nat,set(nat),set_ord_lessThan(nat),N)) = set_or1337092689740270186AtMost(nat,one_one(nat),N) ).

% image_Suc_lessThan
tff(fact_4337_image__Suc__atMost,axiom,
    ! [N: nat] : image2(nat,nat,suc,aa(nat,set(nat),set_ord_atMost(nat),N)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,N)) ).

% image_Suc_atMost
tff(fact_4338_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image2(nat,nat,suc,set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_4339_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image2(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_4340_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image2(nat,nat,suc,aa(nat,set(nat),set_ord_lessThan(nat),N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_4341_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image2(nat,nat,suc,aa(nat,set(nat),set_ord_atMost(nat),N))) ).

% atMost_Suc_eq_insert_0
tff(fact_4342_nths__append,axiom,
    ! [A: $tType,L: list(A),L2: list(A),A6: set(nat)] : nths(A,append(A,L,L2),A6) = append(A,nths(A,L,A6),nths(A,L2,aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_kx(list(A),fun(set(nat),fun(nat,bool)),L),A6)))) ).

% nths_append
tff(fact_4343_length__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),nat,size_size(list(A)),nths(A,Xs,I5)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_ky(list(A),fun(set(nat),fun(nat,bool)),Xs),I5))) ).

% length_nths
tff(fact_4344_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => ( image2(A,A,aa(A,fun(A,A),times_times(A),C3),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),X),aa(A,A,aa(A,fun(A,A),times_times(A),C3),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => ( image2(A,A,aa(A,fun(A,A),times_times(A),C3),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C3),X)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => ( image2(A,A,aa(A,fun(A,A),times_times(A),C3),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_4345_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,C3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
               => ( image2(A,A,aTP_Lamp_kz(A,fun(A,A),C3),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C3),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C3)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
               => ( image2(A,A,aTP_Lamp_kz(A,fun(A,A),C3),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C3),aa(A,A,aa(A,fun(A,A),times_times(A),X),C3)) ) ) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( image2(A,A,aTP_Lamp_kz(A,fun(A,A),C3),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_4346_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M2: A,C3: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_la(A,fun(A,fun(A,A)),M2),C3),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_la(A,fun(A,fun(A,A)),M2),C3),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2)),C3)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_la(A,fun(A,fun(A,A)),M2),C3),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C3)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_4347_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M2: A,C3: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_lb(A,fun(A,fun(A,A)),M2),C3),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_lb(A,fun(A,fun(A,A)),M2),C3),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2)),C3)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_lb(A,fun(A,fun(A,A)),M2),C3),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),B2)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M2),A3)),C3)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_4348_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M2: A,C3: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_lc(A,fun(A,fun(A,A)),M2),C3),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_lc(A,fun(A,fun(A,A)),M2),C3),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,M2)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,M2)),C3)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_lc(A,fun(A,fun(A,A)),M2),C3),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,M2)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,M2)),C3)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_4349_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,M2: A,C3: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
           => ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_ld(A,fun(A,fun(A,A)),M2),C3),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_ld(A,fun(A,fun(A,A)),M2),C3),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,M2)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,M2)),C3)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M2))
               => ( image2(A,A,aa(A,fun(A,A),aTP_Lamp_ld(A,fun(A,fun(A,A)),M2),C3),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,M2)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A3,M2)),C3)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_4350_listrelp__listrel__eq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),X5: list(A),Xa: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X5),Xa))
    <=> pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),X5),Xa)),listrel(A,B,R))) ) ).

% listrelp_listrel_eq
tff(fact_4351_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S3: set(A),R2: set(B),G3: fun(A,B),F3: fun(B,C)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S3))
         => ( pp(aa(set(B),bool,finite_finite2(B),R2))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image2(A,B,G3,S3)),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_le(fun(A,B),fun(fun(B,C),fun(A,C)),G3),F3)),S3) = 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_lg(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S3),G3),F3)),R2) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_4352_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(A)] : image2(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),S3) = S3 ) ).

% image_add_0
tff(fact_4353_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2)))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I2,J2))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_4354_uminus__int__def,axiom,
    uminus_uminus(int) = aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_kl(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_4355_rat__inverse__code,axiom,
    ! [P: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P)) = 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_lh(int,fun(int,product_prod(int,int)))),quotient_of(P)) ).

% rat_inverse_code
tff(fact_4356_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A6: set(product_prod(A,B)),F3: fun(A,fun(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),A6))
     => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,aa(A,fun(B,C),F3,A3),B2)),image2(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),A6))) ) ).

% pair_imageI
tff(fact_4357_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa(set(nat),bool,finite_finite2(nat),set_or3652927894154168847AtMost(nat,L,U))) ).

% finite_greaterThanAtMost
tff(fact_4358_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),set_or3652927894154168847AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),U)) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_4359_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K2))
         => ( set_or3652927894154168847AtMost(A,K2,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_4360_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K2: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K2,L) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),L)) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_4361_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K2: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K2,L) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),L)) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_4362_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),set_or3652927894154168847AtMost(A,A3,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).

% infinite_Ioc_iff
tff(fact_4363_image__add__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [C3: A,A3: A,B2: A] : image2(A,A,aa(A,fun(A,A),plus_plus(A),C3),set_or3652927894154168847AtMost(A,A3,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ).

% image_add_greaterThanAtMost
tff(fact_4364_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or3652927894154168847AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ).

% card_greaterThanAtMost
tff(fact_4365_image__minus__const__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A3: A,B2: A] : image2(A,A,aa(A,fun(A,A),minus_minus(A),C3),set_or3652927894154168847AtMost(A,A3,B2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A3)) ) ).

% image_minus_const_greaterThanAtMost
tff(fact_4366_image__diff__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A3: A,B2: A] : image2(A,A,aa(A,fun(A,A),minus_minus(A),C3),set_or7035219750837199246ssThan(A,A3,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A3)) ) ).

% image_diff_atLeastLessThan
tff(fact_4367_image__uminus__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A] : image2(A,A,uminus_uminus(A),set_or7035219750837199246ssThan(A,X,Y)) = set_or3652927894154168847AtMost(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_atLeastLessThan
tff(fact_4368_image__uminus__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A] : image2(A,A,uminus_uminus(A),set_or3652927894154168847AtMost(A,X,Y)) = set_or7035219750837199246ssThan(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_greaterThanAtMost
tff(fact_4369_divide__rat__def,axiom,
    ! [Q2: rat,R: rat] : divide_divide(rat,Q2,R) = aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),Q2),aa(rat,rat,inverse_inverse(rat),R)) ).

% divide_rat_def
tff(fact_4370_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( set_or3652927894154168847AtMost(A,A3,B2) = set_or3652927894154168847AtMost(A,C3,D3) )
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C3)) )
            | ( ( A3 = C3 )
              & ( B2 = D3 ) ) ) ) ) ).

% Ioc_inj
tff(fact_4371_rat__less__eq__code,axiom,
    ! [P: rat,Q2: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),P),Q2))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_lj(rat,fun(int,fun(int,bool)),Q2)),quotient_of(P))) ) ).

% rat_less_eq_code
tff(fact_4372_rat__less__code,axiom,
    ! [P: rat,Q2: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),P),Q2))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_ll(rat,fun(int,fun(int,bool)),Q2)),quotient_of(P))) ) ).

% rat_less_code
tff(fact_4373_None__notin__image__Some,axiom,
    ! [A: $tType,A6: set(A)] : ~ pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),none(A)),image2(A,option(A),some(A),A6))) ).

% None_notin_image_Some
tff(fact_4374_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,L),U) = set_or3652927894154168847AtMost(nat,L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_4375_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_4376_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ~ pp(aa(set(A),bool,finite_finite2(A),set_or3652927894154168847AtMost(A,A3,B2))) ) ) ).

% infinite_Ioc
tff(fact_4377_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or3652927894154168847AtMost(A,M2,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_4378_ivl__disj__int__two_I6_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or3652927894154168847AtMost(A,M2,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(6)
tff(fact_4379_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C3,D3)) = bot_bot(set(A)) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C3))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),A3)) ) ) ) ).

% Ioc_disjoint
tff(fact_4380_image__int__atLeastAtMost,axiom,
    ! [A3: nat,B2: nat] : image2(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,A3,B2)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastAtMost
tff(fact_4381_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or3652927894154168847AtMost(A,M2,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_4382_ivl__disj__int__two_I8_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M2)),set_or3652927894154168847AtMost(A,M2,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(8)
tff(fact_4383_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),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_4384_ivl__disj__int__one_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(3)
tff(fact_4385_ivl__disj__int__two_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M2: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or5935395276787703475ssThan(A,M2,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(2)
tff(fact_4386_image__int__atLeastLessThan,axiom,
    ! [A3: nat,B2: nat] : image2(nat,int,semiring_1_of_nat(int),set_or7035219750837199246ssThan(nat,A3,B2)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastLessThan
tff(fact_4387_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or3652927894154168847AtMost(nat,M2,N))) ) ) ) ).

% sum.head
tff(fact_4388_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X6: set(A),A6: set(product_prod(A,B)),Y6: set(B),P2: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool))] :
      ( ( X6 = image2(product_prod(A,B),A,product_fst(A,B),A6) )
     => ( ( Y6 = image2(product_prod(A,B),B,product_snd(A,B),A6) )
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
             => ! [Xa4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa4),Y6))
                 => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,X3),Xa4))
                   => pp(aa(B,bool,aa(A,fun(B,bool),Q,X3),Xa4)) ) ) )
         => ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2))))
           => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Q)))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_4389_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or3652927894154168847AtMost(nat,M2,N))) ) ) ) ).

% prod.head
tff(fact_4390_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_4391_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or7035219750837199246ssThan(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D3)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_4392_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_4393_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or3652927894154168847AtMost(A,C3,D3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_4394_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B2: A] : set_or3652927894154168847AtMost(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_4395_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or5935395276787703475ssThan(A,M2,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_4396_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M2)),set_or7035219750837199246ssThan(A,M2,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_4397_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,I2)),J2))
     => ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I2,J2)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I2)),linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I2),J2))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_4398_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image2(int,int,aTP_Lamp_lm(int,fun(int,int),L),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_4399_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_4400_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M2: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M2)),set_or1337092689740270186AtMost(A,M2,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_4401_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),insert(A,U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_4402_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),U))
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = image2(nat,int,semiring_1_of_nat(int),aa(nat,set(nat),set_ord_lessThan(nat),nat2(U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_4403_times__int__def,axiom,
    times_times(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_4404_minus__int__def,axiom,
    minus_minus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_4405_plus__int__def,axiom,
    plus_plus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_4406_rat__minus__code,axiom,
    ! [P: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P),Q2)) = 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_lo(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P)) ).

% rat_minus_code
tff(fact_4407_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa(set(int),bool,finite_finite2(int),set_or3652927894154168847AtMost(int,L,U))) ).

% finite_greaterThanAtMost_int
tff(fact_4408_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or3652927894154168847AtMost(int,L,U)) = nat2(aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ).

% card_greaterThanAtMost_int
tff(fact_4409_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or3652927894154168847AtMost(int,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_4410_normalize__crossproduct,axiom,
    ! [Q2: int,S2: int,P: int,R: int] :
      ( ( Q2 != zero_zero(int) )
     => ( ( S2 != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R),S2)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P),S2) = aa(int,int,aa(int,fun(int,int),times_times(int),R),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_4411_rat__times__code,axiom,
    ! [P: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P),Q2)) = 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_lq(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P)) ).

% rat_times_code
tff(fact_4412_rat__divide__code,axiom,
    ! [P: rat,Q2: rat] : quotient_of(divide_divide(rat,P,Q2)) = 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_ls(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P)) ).

% rat_divide_code
tff(fact_4413_rat__plus__code,axiom,
    ! [P: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P),Q2)) = 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_lu(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P)) ).

% rat_plus_code
tff(fact_4414_div__add__self1__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A3: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( B2 != zero_zero(A) )
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,B2)),one_one(A)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_4415_div__add__self2__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A3: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( B2 != zero_zero(A) )
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,B2)),one_one(A)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_4416_Some__image__these__eq,axiom,
    ! [A: $tType,A6: set(option(A))] : image2(A,option(A),some(A),these(A,A6)) = aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_lv(set(option(A)),fun(option(A),bool),A6)) ).

% Some_image_these_eq
tff(fact_4417_nth__image,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( image2(nat,A,nth(A,Xs),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = aa(list(A),set(A),set2(A),take(A,L,Xs)) ) ) ).

% nth_image
tff(fact_4418_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_4419_these__insert__None,axiom,
    ! [A: $tType,A6: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert(option(A),none(A)),A6)) = these(A,A6) ).

% these_insert_None
tff(fact_4420_take__Suc__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : take(A,aa(nat,nat,suc,N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),take(A,N,Xs)) ).

% take_Suc_Cons
tff(fact_4421_take__eq__Nil2,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( nil(A) = take(A,N,Xs) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil2
tff(fact_4422_take__eq__Nil,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( take(A,N,Xs) = nil(A) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil
tff(fact_4423_take0,axiom,
    ! [A: $tType,X5: list(A)] : take(A,zero_zero(nat),X5) = nil(A) ).

% take0
tff(fact_4424_take__all__iff,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( take(A,N,Xs) = Xs )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% take_all_iff
tff(fact_4425_take__all,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
     => ( take(A,N,Xs) = Xs ) ) ).

% take_all
tff(fact_4426_take__update__cancel,axiom,
    ! [A: $tType,N: nat,M2: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( take(A,N,list_update(A,Xs,M2,Y)) = take(A,N,Xs) ) ) ).

% take_update_cancel
tff(fact_4427_these__insert__Some,axiom,
    ! [A: $tType,X: A,A6: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert(option(A),aa(A,option(A),some(A),X)),A6)) = aa(set(A),set(A),insert(A,X),these(A,A6)) ).

% these_insert_Some
tff(fact_4428_these__image__Some__eq,axiom,
    ! [A: $tType,A6: set(A)] : these(A,image2(A,option(A),some(A),A6)) = A6 ).

% these_image_Some_eq
tff(fact_4429_take__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : take(A,N,append(A,Xs,Ys)) = append(A,take(A,N,Xs),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% take_append
tff(fact_4430_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : take(A,zero_zero(nat),Xs) = nil(A) ).

% take_0
tff(fact_4431_in__these__eq,axiom,
    ! [A: $tType,X: A,A6: set(option(A))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),these(A,A6)))
    <=> pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),aa(A,option(A),some(A),X)),A6)) ) ).

% in_these_eq
tff(fact_4432_set__take__subset__set__take,axiom,
    ! [A: $tType,M2: nat,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,M2,Xs))),aa(list(A),set(A),set2(A),take(A,N,Xs)))) ) ).

% set_take_subset_set_take
tff(fact_4433_nth__take__lemma,axiom,
    ! [A: $tType,K2: nat,Xs: list(A),Ys: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),aa(list(A),nat,size_size(list(A)),Ys)))
       => ( ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),K2))
             => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),I3) ) )
         => ( take(A,K2,Xs) = take(A,K2,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_4434_these__not__empty__eq,axiom,
    ! [A: $tType,B6: set(option(A))] :
      ( ( these(A,B6) != bot_bot(set(A)) )
    <=> ( ( B6 != bot_bot(set(option(A))) )
        & ( B6 != aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_4435_these__empty__eq,axiom,
    ! [A: $tType,B6: set(option(A))] :
      ( ( these(A,B6) = bot_bot(set(A)) )
    <=> ( ( B6 = bot_bot(set(option(A))) )
        | ( B6 = aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_4436_distrib__right__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y: B,C3: A,A3: A,B2: A] :
          ( nO_MATCH(B,A,divide_divide(B,X,Y),C3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4437_distrib__left__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y: B,A3: A,B2: A,C3: A] :
          ( nO_MATCH(B,A,divide_divide(B,X,Y),A3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4438_right__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y: B,A3: A,B2: A,C3: A] :
          ( nO_MATCH(B,A,divide_divide(B,X,Y),A3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4439_left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y: B,C3: A,A3: A,B2: A] :
          ( nO_MATCH(B,A,divide_divide(B,X,Y),C3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4440_Option_Othese__def,axiom,
    ! [A: $tType,A6: set(option(A))] : these(A,A6) = image2(option(A),A,the2(A),aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_lv(set(option(A)),fun(option(A),bool),A6))) ).

% Option.these_def
tff(fact_4441_take__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = nil(A) ) )
      & ( ( N != zero_zero(nat) )
       => ( take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs)) ) ) ) ).

% take_Cons'
tff(fact_4442_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,N: nat] :
          ( nO_MATCH(A,A,one_one(A),X)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) ) ) ) ).

% power_minus'
tff(fact_4443_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R)))
     => ~ ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Ys)))
             => ( ( take(A,I3,Xs) = take(A,I3,Ys) )
               => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Ys),I3))),R)) ) ) ) ) ).

% lex_take_index
tff(fact_4444_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( take(A,aa(nat,nat,suc,I2),Xs) = append(A,take(A,I2,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_4445_nth__repl,axiom,
    ! [A: $tType,M2: nat,Xs: list(A),N: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( ( M2 != N )
         => ( aa(nat,A,nth(A,append(A,take(A,N,Xs),append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),Xs)))),M2) = aa(nat,A,nth(A,Xs),M2) ) ) ) ) ).

% nth_repl
tff(fact_4446_pos__n__replace,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),append(A,take(A,N,Xs),append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)),drop(A,aa(nat,nat,suc,N),Xs)))) ) ) ).

% pos_n_replace
tff(fact_4447_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),A3: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( list_update(A,Xs,I2,A3) = append(A,take(A,I2,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),drop(A,aa(nat,nat,suc,I2),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_4448_id__take__nth__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( Xs = append(A,take(A,I2,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),drop(A,aa(nat,nat,suc,I2),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_4449_drop0,axiom,
    ! [A: $tType,X5: list(A)] : drop(A,zero_zero(nat),X5) = X5 ).

% drop0
tff(fact_4450_drop__Suc__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : drop(A,aa(nat,nat,suc,N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = drop(A,N,Xs) ).

% drop_Suc_Cons
tff(fact_4451_length__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),drop(A,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_drop
tff(fact_4452_drop__eq__Nil2,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( nil(A) = drop(A,N,Xs) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_4453_drop__eq__Nil,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( drop(A,N,Xs) = nil(A) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_4454_drop__all,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
     => ( drop(A,N,Xs) = nil(A) ) ) ).

% drop_all
tff(fact_4455_drop__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : drop(A,N,append(A,Xs,Ys)) = append(A,drop(A,N,Xs),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% drop_append
tff(fact_4456_nth__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A),I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,drop(A,N,Xs)),I2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),I2)) ) ) ).

% nth_drop
tff(fact_4457_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : drop(A,zero_zero(nat),Xs) = Xs ).

% drop_0
tff(fact_4458_drop__eq__nths,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,N,Xs) = nths(A,Xs,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),ord_less_eq(nat),N))) ).

% drop_eq_nths
tff(fact_4459_set__drop__subset__set__drop,axiom,
    ! [A: $tType,N: nat,M2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,M2,Xs))),aa(list(A),set(A),set2(A),drop(A,N,Xs)))) ) ).

% set_drop_subset_set_drop
tff(fact_4460_drop__update__swap,axiom,
    ! [A: $tType,M2: nat,N: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( drop(A,M2,list_update(A,Xs,N,X)) = list_update(A,drop(A,M2,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2),X) ) ) ).

% drop_update_swap
tff(fact_4461_append__eq__conv__conj,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( append(A,Xs,Ys) = Zs )
    <=> ( ( Xs = take(A,aa(list(A),nat,size_size(list(A)),Xs),Zs) )
        & ( Ys = drop(A,aa(list(A),nat,size_size(list(A)),Xs),Zs) ) ) ) ).

% append_eq_conv_conj
tff(fact_4462_drop__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
      & ( ( N != zero_zero(nat) )
       => ( drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs) ) ) ) ).

% drop_Cons'
tff(fact_4463_append__eq__append__conv__if,axiom,
    ! [A: $tType,Xs_1: list(A),Xs_2: list(A),Ys_1: list(A),Ys_2: list(A)] :
      ( ( append(A,Xs_1,Xs_2) = append(A,Ys_1,Ys_2) )
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
         => ( ( Xs_1 = take(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1) )
            & ( Xs_2 = append(A,drop(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1),Ys_2) ) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
         => ( ( take(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1) = Ys_1 )
            & ( append(A,drop(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1),Xs_2) = Ys_2 ) ) ) ) ) ).

% append_eq_append_conv_if
tff(fact_4464_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),drop(A,aa(nat,nat,suc,I2),Xs)) = drop(A,I2,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_4465_zip__append2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(B)] : zip(A,B,Xs,append(B,Ys,Zs)) = append(product_prod(A,B),zip(A,B,take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs),Ys),zip(A,B,drop(A,aa(list(B),nat,size_size(list(B)),Ys),Xs),Zs)) ).

% zip_append2
tff(fact_4466_zip__append1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(A),Zs: list(B)] : zip(A,B,append(A,Xs,Ys),Zs) = append(product_prod(A,B),zip(A,B,Xs,take(B,aa(list(A),nat,size_size(list(A)),Xs),Zs)),zip(A,B,Ys,drop(B,aa(list(A),nat,size_size(list(A)),Xs),Zs))) ).

% zip_append1
tff(fact_4467_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),I2: nat,J2: nat] :
      ( distinct(A,Vs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),take(A,I2,Vs))),aa(list(A),set(A),set2(A),drop(A,J2,Vs))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_4468_take__hd__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( append(A,take(A,N,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),hd(A,drop(A,N,Xs))),nil(A))) = take(A,aa(nat,nat,suc,N),Xs) ) ) ).

% take_hd_drop
tff(fact_4469_rotate__drop__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,N),Xs) = append(A,drop(A,modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)),Xs),take(A,modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)),Xs)) ).

% rotate_drop_take
tff(fact_4470_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_lw(A,fun(A,bool))) = ord_max(A) ) ) ).

% dual_min
tff(fact_4471_list__encode_Opelims,axiom,
    ! [X: list(nat),Y: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,X) = Y )
     => ( accp(list(nat),nat_list_encode_rel,X)
       => ( ( ( X = nil(nat) )
           => ( ( Y = zero_zero(nat) )
             => ~ accp(list(nat),nat_list_encode_rel,nil(nat)) ) )
         => ~ ! [X3: nat,Xs2: list(nat)] :
                ( ( X = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),Xs2) )
               => ( ( Y = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),aa(list(nat),nat,nat_list_encode,Xs2)))) )
                 => ~ accp(list(nat),nat_list_encode_rel,aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_4472_length__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rotate(A,N),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rotate
tff(fact_4473_hd__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( hd(A,replicate(A,N,X)) = X ) ) ).

% hd_replicate
tff(fact_4474_rotate__Suc,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,aa(nat,nat,suc,N)),Xs) = rotate1(A,aa(list(A),list(A),rotate(A,N),Xs)) ).

% rotate_Suc
tff(fact_4475_hd__take,axiom,
    ! [A: $tType,J2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),J2))
     => ( hd(A,take(A,J2,Xs)) = hd(A,Xs) ) ) ).

% hd_take
tff(fact_4476_rotate__length01,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( aa(list(A),list(A),rotate(A,N),Xs) = Xs ) ) ).

% rotate_length01
tff(fact_4477_rotate__id,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)) = zero_zero(nat) )
     => ( aa(list(A),list(A),rotate(A,N),Xs) = Xs ) ) ).

% rotate_id
tff(fact_4478_hd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( hd(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),hd(A,Xs)),hd(B,Ys)) ) ) ) ).

% hd_zip
tff(fact_4479_ord_Omin__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A3),B2) = A3 ) )
      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A3),B2) = B2 ) ) ) ).

% ord.min_def
tff(fact_4480_ord_Omin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : min(A,Less_eq) = min(A,Less_eq) ).

% ord.min.cong
tff(fact_4481_hd__rotate__conv__nth,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( Xs != nil(A) )
     => ( hd(A,aa(list(A),list(A),rotate(A,N),Xs)) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% hd_rotate_conv_nth
tff(fact_4482_rotate__append,axiom,
    ! [A: $tType,L: list(A),Q2: list(A)] : aa(list(A),list(A),rotate(A,aa(list(A),nat,size_size(list(A)),L)),append(A,L,Q2)) = append(A,Q2,L) ).

% rotate_append
tff(fact_4483_rotate__conv__mod,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,N),Xs) = aa(list(A),list(A),rotate(A,modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% rotate_conv_mod
tff(fact_4484_hd__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( hd(A,Xs) = aa(nat,A,nth(A,Xs),zero_zero(nat)) ) ) ).

% hd_conv_nth
tff(fact_4485_hd__drop__conv__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( hd(A,drop(A,N,Xs)) = aa(nat,A,nth(A,Xs),N) ) ) ).

% hd_drop_conv_nth
tff(fact_4486_nth__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A),M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate(A,M2),Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate
tff(fact_4487_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,nat)] :
      ( ( ( Xs = nil(A) )
       => ( size_list(A,F3,Xs) = zero_zero(nat) ) )
      & ( ( Xs != nil(A) )
       => ( size_list(A,F3,Xs) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,hd(A,Xs))),size_list(A,F3,tl(A,Xs)))) ) ) ) ).

% Nitpick.size_list_simp(1)
tff(fact_4488_card__Min__le__sum,axiom,
    ! [A: $tType,A6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),image2(A,nat,F3,A6)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A6))) ) ).

% card_Min_le_sum
tff(fact_4489_remdups__adj__singleton__iff,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Xs != nil(A) )
        & ( Xs = replicate(A,aa(list(A),nat,size_size(list(A)),Xs),hd(A,Xs)) ) ) ) ).

% remdups_adj_singleton_iff
tff(fact_4490_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M2)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M2),aa(num,nat,numeral_numeral(nat),N)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(num,nat,numeral_numeral(nat),M2)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M2)),aa(num,nat,numeral_numeral(nat),N))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M2)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M2),aa(num,nat,numeral_numeral(nat),N)) = nil(nat) ) ) ) ).

% upt_rec_numeral
tff(fact_4491_tl__upt,axiom,
    ! [M2: nat,N: nat] : tl(nat,upt(M2,N)) = upt(aa(nat,nat,suc,M2),N) ).

% tl_upt
tff(fact_4492_length__upt,axiom,
    ! [I2: nat,J2: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I2,J2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J2),I2) ).

% length_upt
tff(fact_4493_take__upt,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M2)),N))
     => ( take(nat,M2,upt(I2,N)) = upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M2)) ) ) ).

% take_upt
tff(fact_4494_upt__conv__Nil,axiom,
    ! [J2: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
     => ( upt(I2,J2) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_4495_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A6)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_4496_length__tl,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),tl(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).

% length_tl
tff(fact_4497_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J2: nat] :
      ( ( upt(I2,J2) = nil(nat) )
    <=> ( ( J2 = zero_zero(nat) )
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2)) ) ) ).

% upt_eq_Nil_conv
tff(fact_4498_atLeastAtMost__upt,axiom,
    ! [N: nat,M2: nat] : set_or1337092689740270186AtMost(nat,N,M2) = aa(list(nat),set(nat),set2(nat),upt(N,aa(nat,nat,suc,M2))) ).

% atLeastAtMost_upt
tff(fact_4499_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A6)),A3)) ) ) ) ).

% Min.coboundedI
tff(fact_4500_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ! [Y3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A6) = X ) ) ) ) ) ).

% Min_eqI
tff(fact_4501_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A6)),X)) ) ) ) ).

% Min_le
tff(fact_4502_atLeast__upt,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),N)) ).

% atLeast_upt
tff(fact_4503_remdups__adj__length,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% remdups_adj_length
tff(fact_4504_take__tl,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : take(A,N,tl(A,Xs)) = tl(A,take(A,aa(nat,nat,suc,N),Xs)) ).

% take_tl
tff(fact_4505_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list(nat),Q2: nat] :
      ( ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),M2),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns)) = upt(M2,Q2) )
    <=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns) = upt(aa(nat,nat,suc,M2),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_4506_drop__Suc,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,aa(nat,nat,suc,N),Xs) = drop(A,N,tl(A,Xs)) ).

% drop_Suc
tff(fact_4507_upt__0,axiom,
    ! [I2: nat] : upt(I2,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_4508_greaterThanAtMost__upt,axiom,
    ! [N: nat,M2: nat] : set_or3652927894154168847AtMost(nat,N,M2) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),aa(nat,nat,suc,M2))) ).

% greaterThanAtMost_upt
tff(fact_4509_greaterThanLessThan__upt,axiom,
    ! [N: nat,M2: nat] : set_or5935395276787703475ssThan(nat,N,M2) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),M2)) ).

% greaterThanLessThan_upt
tff(fact_4510_atMost__upto,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,N))) ).

% atMost_upto
tff(fact_4511_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A5)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A6))) ) ) ) ) ).

% Min.boundedI
tff(fact_4512_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A6)))
             => ! [A8: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A8),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A8)) ) ) ) ) ) ).

% Min.boundedE
tff(fact_4513_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),M2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ( M2 = aa(set(A),A,lattic643756798350308766er_Min(A),A6) )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M2),A6))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),X4)) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_4514_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A6)),X))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).

% Min_le_iff
tff(fact_4515_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),M2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A6) = M2 )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M2),A6))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),X4)) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_4516_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ! [B4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B4)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),insert(A,A3),A6)) = A3 ) ) ) ) ).

% Min_insert2
tff(fact_4517_upt__conv__Cons,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
     => ( upt(I2,J2) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),upt(aa(nat,nat,suc,I2),J2)) ) ) ).

% upt_conv_Cons
tff(fact_4518_enumerate__eq__zip,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : enumerate(A,N,Xs) = zip(nat,A,upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% enumerate_eq_zip
tff(fact_4519_upt__add__eq__append,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2)) = append(nat,upt(I2,J2),upt(J2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),K2))) ) ) ).

% upt_add_eq_append
tff(fact_4520_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),B6: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B6)),aa(set(A),A,lattic643756798350308766er_Min(A),A6))) ) ) ) ) ).

% Min.subset_imp
tff(fact_4521_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M5: set(A),N6: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M5),N6))
         => ( ( M5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),N6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N6)),aa(set(A),A,lattic643756798350308766er_Min(A),M5))) ) ) ) ) ).

% Min_antimono
tff(fact_4522_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( ( Xs = nil(A) )
       => ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) ) )
      & ( ( Xs != nil(A) )
       => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),tl(A,Xs))) ) ) ) ).

% Nitpick.size_list_simp(2)
tff(fact_4523_remdups__adj__adjacent,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))))
     => ( aa(nat,A,nth(A,remdups_adj(A,Xs)),I2) != aa(nat,A,nth(A,remdups_adj(A,Xs)),aa(nat,nat,suc,I2)) ) ) ).

% remdups_adj_adjacent
tff(fact_4524_nth__tl,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),tl(A,Xs))))
     => ( aa(nat,A,nth(A,tl(A,Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N)) ) ) ).

% nth_tl
tff(fact_4525_remdups__adj__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( ( N = zero_zero(nat) )
       => ( remdups_adj(A,replicate(A,N,X)) = nil(A) ) )
      & ( ( N != zero_zero(nat) )
       => ( remdups_adj(A,replicate(A,N,X)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ) ) ).

% remdups_adj_replicate
tff(fact_4526_remdups__adj__singleton,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ( remdups_adj(A,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
     => ( Xs = replicate(A,aa(list(A),nat,size_size(list(A)),Xs),X) ) ) ).

% remdups_adj_singleton
tff(fact_4527_upt__rec,axiom,
    ! [I2: nat,J2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
       => ( upt(I2,J2) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),upt(aa(nat,nat,suc,I2),J2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
       => ( upt(I2,J2) = nil(nat) ) ) ) ).

% upt_rec
tff(fact_4528_remdups__adj__length__ge1,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)))) ) ).

% remdups_adj_length_ge1
tff(fact_4529_take__Suc,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( Xs != nil(A) )
     => ( take(A,aa(nat,nat,suc,N),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),hd(A,Xs)),take(A,N,tl(A,Xs))) ) ) ).

% take_Suc
tff(fact_4530_upt__Suc,axiom,
    ! [I2: nat,J2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
       => ( upt(I2,aa(nat,nat,suc,J2)) = append(nat,upt(I2,J2),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J2),nil(nat))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
       => ( upt(I2,aa(nat,nat,suc,J2)) = nil(nat) ) ) ) ).

% upt_Suc
tff(fact_4531_upt__Suc__append,axiom,
    ! [I2: nat,J2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
     => ( upt(I2,aa(nat,nat,suc,J2)) = append(nat,upt(I2,J2),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J2),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_4532_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] : aa(list(bool),A,aa(A,fun(list(bool),A),aa(fun(bool,A),fun(A,fun(list(bool),A)),groups4207007520872428315er_sum(bool,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),bit0(one2))),map(nat,bool,bit_se5641148757651400278ts_bit(A,A3),upt(zero_zero(nat),N))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_4533_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys )
    <=> ? [F5: fun(nat,nat)] :
          ( pp(aa(fun(nat,nat),bool,order_mono(nat,nat),F5))
          & ( image2(nat,nat,F5,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys)) )
          & ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( aa(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Ys),aa(nat,nat,F5,I)) ) )
          & ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( ( aa(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat))) )
              <=> ( aa(nat,nat,F5,I) = aa(nat,nat,F5,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_4534_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_lw(A,fun(A,bool))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_4535_map__fst__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : map(product_prod(nat,A),nat,product_fst(nat,A),enumerate(A,N,Xs)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ).

% map_fst_enumerate
tff(fact_4536_length__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),map(B,A,F3,Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_map
tff(fact_4537_nth__map,axiom,
    ! [B: $tType,A: $tType,N: nat,Xs: list(A),F3: fun(A,B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,B,nth(B,map(A,B,F3,Xs)),N) = aa(A,B,F3,aa(nat,A,nth(A,Xs),N)) ) ) ).

% nth_map
tff(fact_4538_map__fst__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map(product_prod(A,B),A,product_fst(A,B),zip(A,B,Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_4539_map__snd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map(product_prod(A,B),B,product_snd(A,B),zip(A,B,Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_4540_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),Q))
         => pp(aa(fun(nat,A),bool,order_mono(nat,A),aTP_Lamp_lx(fun(A,A),fun(nat,A),Q))) ) ) ).

% mono_funpow
tff(fact_4541_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,map(C,A,F3,Xs),Ys) = map(product_prod(C,B),product_prod(A,B),aa(fun(C,fun(B,product_prod(A,B))),fun(product_prod(C,B),product_prod(A,B)),product_case_prod(C,B,product_prod(A,B)),aTP_Lamp_ly(fun(C,A),fun(C,fun(B,product_prod(A,B))),F3)),zip(C,B,Xs,Ys)) ).

% zip_map1
tff(fact_4542_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F3: fun(C,B),Ys: list(C)] : zip(A,B,Xs,map(C,B,F3,Ys)) = map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_lz(fun(C,B),fun(A,fun(C,product_prod(A,B))),F3)),zip(A,C,Xs,Ys)) ).

% zip_map2
tff(fact_4543_zip__map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,F3: fun(C,A),Xs: list(C),G3: fun(D,B),Ys: list(D)] : zip(A,B,map(C,A,F3,Xs),map(D,B,G3,Ys)) = map(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_ma(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F3),G3)),zip(C,D,Xs,Ys)) ).

% zip_map_map
tff(fact_4544_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(product_prod(B,C),A),G3: fun(D,B),Xs: list(D),Ys: list(C)] : map(product_prod(B,C),A,F3,zip(B,C,map(D,B,G3,Xs),Ys)) = map(product_prod(D,C),A,aa(fun(D,fun(C,A)),fun(product_prod(D,C),A),product_case_prod(D,C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_mb(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F3),G3)),zip(D,C,Xs,Ys)) ).

% map_zip_map
tff(fact_4545_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(product_prod(B,C),A),Xs: list(B),G3: fun(D,C),Ys: list(D)] : map(product_prod(B,C),A,F3,zip(B,C,Xs,map(D,C,G3,Ys))) = map(product_prod(B,D),A,aa(fun(B,fun(D,A)),fun(product_prod(B,D),A),product_case_prod(B,D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_mc(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F3),G3)),zip(B,D,Xs,Ys)) ).

% map_zip_map2
tff(fact_4546_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% mono_strict_invE
tff(fact_4547_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
        <=> ! [X4: A,Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,F3,Y5))) ) ) ) ).

% mono_def
tff(fact_4548_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) )
         => pp(aa(fun(A,B),bool,order_mono(A,B),F3)) ) ) ).

% monoI
tff(fact_4549_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ).

% monoE
tff(fact_4550_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ).

% monoD
tff(fact_4551_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(B,A),Xs: list(B),G3: fun(C,A),Ys: list(C)] :
      ( ( map(B,A,F3,Xs) = map(C,A,G3,Ys) )
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4552_mono__Suc,axiom,
    pp(aa(fun(nat,nat),bool,order_mono(nat,nat),suc)) ).

% mono_Suc
tff(fact_4553_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => pp(aa(fun(A,A),bool,order_mono(A,A),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3))) ) ) ).

% mono_pow
tff(fact_4554_mono__add,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A] : pp(aa(fun(A,A),bool,order_mono(A,A),aa(A,fun(A,A),plus_plus(A),A3))) ) ).

% mono_add
tff(fact_4555_enumerate__map__upt,axiom,
    ! [A: $tType,N: nat,F3: fun(nat,A),M2: nat] : enumerate(A,N,map(nat,A,F3,upt(N,M2))) = map(nat,product_prod(nat,A),aTP_Lamp_md(fun(nat,A),fun(nat,product_prod(nat,A)),F3),upt(N,M2)) ).

% enumerate_map_upt
tff(fact_4556_incseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A6: fun(nat,A),I2: nat] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A6,I2)),aa(nat,A,A6,aa(nat,nat,suc,I2)))) ) ) ).

% incseq_SucD
tff(fact_4557_incseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N3)),aa(nat,A,X6,aa(nat,nat,suc,N3))))
         => pp(aa(fun(nat,A),bool,order_mono(nat,A),X6)) ) ) ).

% incseq_SucI
tff(fact_4558_incseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F3))
        <=> ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N5)),aa(nat,A,F3,aa(nat,nat,suc,N5)))) ) ) ).

% incseq_Suc_iff
tff(fact_4559_incseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),X6))
        <=> ! [M6: nat,N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M6),N5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,M6)),aa(nat,A,X6,N5))) ) ) ) ).

% incseq_def
tff(fact_4560_incseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),I2: nat,J2: nat] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,I2)),aa(nat,A,F3,J2))) ) ) ) ).

% incseqD
tff(fact_4561_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mono_invE
tff(fact_4562_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F3: fun(A,B),A6: A,B6: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),A6),B6))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F3,A6)),aa(A,B,F3,B6)))) ) ) ).

% mono_inf
tff(fact_4563_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F3: fun(A,B),A6: A,B6: A] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F3,A6)),aa(A,B,F3,B6))),aa(A,B,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),A6),B6)))) ) ) ).

% mono_sup
tff(fact_4564_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(A,A),A6: A,B6: A,N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),B6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),A6)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),B6))) ) ) ) ).

% funpow_mono
tff(fact_4565_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K2: A,Lst: list(B)] : map(B,A,aa(A,fun(B,A),aTP_Lamp_kc(A,fun(B,A)),K2),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K2) ).

% map_replicate_const
tff(fact_4566_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(fun(nat,nat),bool,order_mono(nat,nat),aa(nat,fun(nat,nat),times_times(nat),N))) ) ).

% mono_times_nat
tff(fact_4567_mono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => pp(aa(fun(A,A),bool,order_mono(A,A),aa(A,fun(A,A),times_times(A),A3))) ) ) ).

% mono_mult
tff(fact_4568_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F3: fun(A,B),M2: A,N: A,M4: B,N2: B] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( ( image2(A,B,F3,set_or7035219750837199246ssThan(A,M2,N)) = set_or7035219750837199246ssThan(B,M4,N2) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M2),N))
             => ( aa(A,B,F3,M2) = M4 ) ) ) ) ) ).

% mono_image_least
tff(fact_4569_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F3: fun(A,A),P: A,K2: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,P)),P))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),bot_bot(A))),P)) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_4570_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(A,A),I2: nat,J2: nat,X: A,Y: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,F3,X)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),I2),F3),X)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),J2),F3),Y))) ) ) ) ) ) ).

% funpow_mono2
tff(fact_4571_Least__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),S3: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( ? [X5: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),S3))
                & ! [Xa4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),S3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa4)) ) )
           => ( ord_Least(B,aa(set(A),fun(B,bool),aTP_Lamp_me(fun(A,B),fun(set(A),fun(B,bool)),F3),S3)) = aa(A,B,F3,ord_Least(A,aTP_Lamp_mf(set(A),fun(A,bool),S3))) ) ) ) ) ).

% Least_mono
tff(fact_4572_map__replicate__trivial,axiom,
    ! [A: $tType,X: A,I2: nat] : map(nat,A,aTP_Lamp_mg(A,fun(nat,A),X),upt(zero_zero(nat),I2)) = replicate(A,I2,X) ).

% map_replicate_trivial
tff(fact_4573_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [M2: nat,N: nat,F3: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M2),F3),bot_bot(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),bot_bot(A)))) ) ) ) ).

% funpow_decreasing
tff(fact_4574_zip__eq__conv,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(product_prod(A,B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( zip(A,B,Xs,Ys) = Zs )
      <=> ( ( map(product_prod(A,B),A,product_fst(A,B),Zs) = Xs )
          & ( map(product_prod(A,B),B,product_snd(A,B),Zs) = Ys ) ) ) ) ).

% zip_eq_conv
tff(fact_4575_map__upt__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),N: nat] : map(nat,A,F3,upt(zero_zero(nat),aa(nat,nat,suc,N))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,F3,zero_zero(nat))),map(nat,A,aTP_Lamp_mh(fun(nat,A),fun(nat,A),F3),upt(zero_zero(nat),N))) ).

% map_upt_Suc
tff(fact_4576_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : map(nat,A,nth(A,Xs),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).

% map_nth
tff(fact_4577_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2))
     => pp(aa(fun(nat,nat),bool,order_mono(nat,nat),aTP_Lamp_mi(nat,fun(nat,nat),K2))) ) ).

% mono_ge2_power_minus_self
tff(fact_4578_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,M2: nat,F3: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),I3)) ) )
       => ( map(nat,A,F3,upt(M2,N)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_4579_eq__key__imp__eq__value,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),K2: A,V1: B,V22: B] :
      ( distinct(A,map(product_prod(A,B),A,product_fst(A,B),Xs))
     => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V1)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
       => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V22)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
tff(fact_4580_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( N = zero_zero(nat) ) )
     => ( ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I3)) = N ) )
       => ( transpose(A,Xs) = map(nat,list(A),aTP_Lamp_mk(list(list(A)),fun(nat,list(A)),Xs),upt(zero_zero(nat),N)) ) ) ) ).

% transpose_rectangle
tff(fact_4581_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),X6))
       => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F3,Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_ml(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F3)),X6) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_4582_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),image2(nat,A,F3,top_top(set(nat)))))
         => ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F3))
           => ( ! [N3: nat] :
                  ( ( aa(nat,A,F3,N3) = aa(nat,A,F3,aa(nat,nat,suc,N3)) )
                 => ( aa(nat,A,F3,aa(nat,nat,suc,N3)) = aa(nat,A,F3,aa(nat,nat,suc,aa(nat,nat,suc,N3))) ) )
             => ? [N9: nat] :
                  ( ! [N4: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),N9))
                     => ! [M3: nat] :
                          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N9))
                         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N4))
                           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,M3)),aa(nat,A,F3,N4))) ) ) )
                  & ! [N4: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),N4))
                     => ( aa(nat,A,F3,N9) = aa(nat,A,F3,N4) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_4583_top__apply,axiom,
    ! [D: $tType,C: $tType] :
      ( top(C)
     => ! [X: D] : aa(D,C,top_top(fun(D,C)),X) = top_top(C) ) ).

% top_apply
tff(fact_4584_atMost__UNIV__triv,axiom,
    ! [A: $tType] : aa(set(A),set(set(A)),set_ord_atMost(set(A)),top_top(set(A))) = top_top(set(set(A))) ).

% atMost_UNIV_triv
tff(fact_4585_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),top_top(set(option(A)))))
    <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ).

% finite_option_UNIV
tff(fact_4586_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_4587_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_4588_surj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat] :
      ( ( image2(A,A,F3,top_top(set(A))) = top_top(set(A)) )
     => ( image2(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_4589_sum__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( aa(list(A),A,groups8242544230860333062m_list(A),nil(A)) = zero_zero(A) ) ) ).

% sum_list.Nil
tff(fact_4590_sum__list__eq__0__iff,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ns: list(A)] :
          ( ( aa(list(A),A,groups8242544230860333062m_list(A),Ns) = zero_zero(A) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Ns)))
             => ( X4 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_4591_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aTP_Lamp_mm(B,A),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_4592_sum__list__upt,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),upt(M2,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_bu(nat,nat)),set_or7035219750837199246ssThan(nat,M2,N)) ) ) ).

% sum_list_upt
tff(fact_4593_length__concat,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(B),nat,size_size(list(B)),concat(B,Xss)) = aa(list(nat),nat,groups8242544230860333062m_list(nat),map(list(B),nat,size_size(list(B)),Xss)) ).

% length_concat
tff(fact_4594_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_mn(list(B),fun(A,list(product_prod(A,B))),Ys),Xs)) ).

% product_concat_map
tff(fact_4595_not__UNIV__eq__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L2: A,H2: A] : top_top(set(A)) != set_or1337092689740270186AtMost(A,L2,H2) ) ).

% not_UNIV_eq_Icc
tff(fact_4596_atMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] :
          ( ( aa(A,set(A),set_ord_atMost(A),X) = top_top(set(A)) )
        <=> ( X = top_top(A) ) ) ) ).

% atMost_eq_UNIV_iff
tff(fact_4597_not__UNIV__eq__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H2: A] : top_top(set(A)) != aa(A,set(A),set_ord_atMost(A),H2) ) ).

% not_UNIV_eq_Iic
tff(fact_4598_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),top_top(A))) ) ).

% top_greatest
tff(fact_4599_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A3))
        <=> ( A3 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_4600_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A3))
         => ( A3 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_4601_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),A3)) ) ).

% top.extremum_strict
tff(fact_4602_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( ( A3 != top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),top_top(A))) ) ) ).

% top.not_eq_extremum
tff(fact_4603_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( bounded_lattice(A)
     => ! [X: A,Y: A] :
          ( ( set_or1337092689740270186AtMost(A,X,Y) = top_top(set(A)) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_4604_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,N),Xs) = concat(list(A),map(list(A),list(list(A)),aTP_Lamp_mp(list(A),fun(list(A),list(list(A))),Xs),n_lists(A,N,Xs))) ).

% n_lists.simps(2)
tff(fact_4605_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( pp(aa(set(fun(A,B)),bool,finite_finite2(fun(A,B)),top_top(set(fun(A,B)))))
     => ( ( aa(set(B),nat,finite_card(B),top_top(set(B))) != aa(nat,nat,suc,zero_zero(nat)) )
       => pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ) ).

% finite_fun_UNIVD1
tff(fact_4606_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] : map(nat,nat,suc,upt(M2,N)) = upt(aa(nat,nat,suc,M2),aa(nat,nat,suc,N)) ).

% map_Suc_upt
tff(fact_4607_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(list(A),A,groups8242544230860333062m_list(A),Xs))) ) ) ).

% member_le_sum_list
tff(fact_4608_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H))) ) ).

% not_UNIV_le_Icc
tff(fact_4609_zip__same__conv__map,axiom,
    ! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = map(A,product_prod(A,A),aTP_Lamp_mq(A,product_prod(A,A)),Xs) ).

% zip_same_conv_map
tff(fact_4610_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atMost(A),H))) ) ).

% not_UNIV_le_Iic
tff(fact_4611_bij__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat] :
      ( bij_betw(A,A,F3,top_top(set(A)),top_top(set(A)))
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_4612_sum__list__mult__const,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [F3: fun(B,A),C3: A,Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_bc(fun(B,A),fun(A,fun(B,A)),F3),C3),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F3,Xs))),C3) ) ).

% sum_list_mult_const
tff(fact_4613_sum__list__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C3: A,F3: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bb(A,fun(fun(B,A),fun(B,A)),C3),F3),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F3,Xs))) ) ).

% sum_list_const_mult
tff(fact_4614_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),set(option(A)),insert(option(A),none(A)),image2(A,option(A),some(A),top_top(set(A)))) ).

% UNIV_option_conv
tff(fact_4615_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% sup_shunt
tff(fact_4616_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F3: fun(A,A),P: A,K2: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P),aa(A,A,F3,P)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),top_top(A)))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_4617_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),zero_zero(A))) ) ) ).

% sum_list_nonpos
tff(fact_4618_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3)) )
         => ( ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = zero_zero(A) )
          <=> ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
               => ( X4 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_4619_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(list(A),A,groups8242544230860333062m_list(A),Xs))) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_4620_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xs: list(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),map(A,A,abs_abs(A),Xs)))) ) ).

% sum_list_abs
tff(fact_4621_map__add__upt,axiom,
    ! [N: nat,M2: nat] : map(nat,nat,aTP_Lamp_mr(nat,fun(nat,nat),N),upt(zero_zero(nat),M2)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)) ).

% map_add_upt
tff(fact_4622_sum__list__replicate,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat,C3: A] : aa(list(A),A,groups8242544230860333062m_list(A),replicate(A,N,C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),C3) ) ).

% sum_list_replicate
tff(fact_4623_zip__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = map(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C)),aa(fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C))),aa(fun(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_ms(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))))),zip(product_prod(A,B),C,zip(A,B,Xs,Ys),Zs)) ).

% zip_assoc
tff(fact_4624_zip__left__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = map(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C)),aa(fun(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_mu(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))))),zip(B,product_prod(A,C),Ys,zip(A,C,Xs,Zs))) ).

% zip_left_commute
tff(fact_4625_zip__commute,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : zip(A,B,Xs,Ys) = map(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_mv(B,fun(A,product_prod(A,B)))),zip(B,A,Ys,Xs)) ).

% zip_commute
tff(fact_4626_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3))) )
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F3,Xs))),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,G3,Xs)))) ) ) ).

% sum_list_mono
tff(fact_4627_finite__range__Some,axiom,
    ! [A: $tType] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),image2(A,option(A),some(A),top_top(set(A)))))
    <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ).

% finite_range_Some
tff(fact_4628_notin__range__Some,axiom,
    ! [A: $tType,X: option(A)] :
      ( ~ pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),X),image2(A,option(A),some(A),top_top(set(A)))))
    <=> ( X = none(A) ) ) ).

% notin_range_Some
tff(fact_4629_zip__replicate1,axiom,
    ! [A: $tType,B: $tType,N: nat,X: A,Ys: list(B)] : zip(A,B,replicate(A,N,X),Ys) = map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),take(B,N,Ys)) ).

% zip_replicate1
tff(fact_4630_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [M2: nat,N: nat,F3: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),top_top(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M2),F3),top_top(A)))) ) ) ) ).

% funpow_increasing
tff(fact_4631_map__decr__upt,axiom,
    ! [M2: nat,N: nat] : map(nat,nat,aTP_Lamp_mw(nat,nat),upt(aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = upt(M2,N) ).

% map_decr_upt
tff(fact_4632_inf__top_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => semila1105856199041335345_order(A,inf_inf(A),top_top(A),ord_less_eq(A),ord_less(A)) ) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_4633_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K2: nat,Ns: list(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(list(A),nat,size_size(list(A)),Ns)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Ns),K2)),aa(list(A),A,groups8242544230860333062m_list(A),Ns))) ) ) ).

% elem_le_sum_list
tff(fact_4634_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),N: nat,Y: B] : zip(A,B,Xs,replicate(B,N,Y)) = map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_mv(B,fun(A,product_prod(A,B))),Y),take(A,N,Xs)) ).

% zip_replicate2
tff(fact_4635_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( pp(aa(set(A),bool,finite_finite2(A),top_top(set(A))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A))))) ) ).

% finite_UNIV_card_ge_0
tff(fact_4636_size__list__conv__sum__list,axiom,
    ! [B: $tType,F3: fun(B,nat),Xs: list(B)] : size_list(B,F3,Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(B,nat,F3,Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% size_list_conv_sum_list
tff(fact_4637_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = append(product_prod(A,B),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Ys),product(A,B,Xs,Ys)) ).

% product.simps(2)
tff(fact_4638_sum__list__triv,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [R: B,Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),map(C,B,aTP_Lamp_mx(B,fun(C,B),R),Xs)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(list(C),nat,size_size(list(C)),Xs))),R) ) ).

% sum_list_triv
tff(fact_4639_sum__list__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,aTP_Lamp_cf(fun(A,nat),fun(A,nat),F3),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F3,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_4640_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),image2(B,A,F3,top_top(set(B)))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),image2(B,A,F3,top_top(set(B)))))) ) ).

% card_range_greater_zero
tff(fact_4641_enumerate__replicate__eq,axiom,
    ! [A: $tType,N: nat,M2: nat,A3: A] : enumerate(A,N,replicate(A,M2,A3)) = map(nat,product_prod(nat,A),aTP_Lamp_my(A,fun(nat,product_prod(nat,A)),A3),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2))) ).

% enumerate_replicate_eq
tff(fact_4642_sum__list__sum__nth,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(B)] : aa(list(B),B,groups8242544230860333062m_list(B),Xs) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),nth(B,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_4643_card__length__sum__list__rec,axiom,
    ! [M2: nat,N6: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M2))
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_mz(nat,fun(nat,fun(list(nat),bool)),M2),N6))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_na(nat,fun(nat,fun(list(nat),bool)),M2),N6)))),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_nb(nat,fun(nat,fun(list(nat),bool)),M2),N6)))) ) ) ).

% card_length_sum_list_rec
tff(fact_4644_card__length__sum__list,axiom,
    ! [M2: nat,N6: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_mz(nat,fun(nat,fun(list(nat),bool)),M2),N6))) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N6),M2)),one_one(nat))),N6) ).

% card_length_sum_list
tff(fact_4645_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F3,Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_nc(fun(A,nat),fun(list(A),fun(A,nat)),F3),Xs)),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_4646_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( image2(nat,nat,aTP_Lamp_nd(nat,fun(nat,nat),N),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).

% range_mod
tff(fact_4647_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K2: nat,Xs: list(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(list(A),nat,size_size(list(A)),Xs)))
         => ( aa(list(A),A,groups8242544230860333062m_list(A),list_update(A,Xs,K2,X)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),X)),aa(nat,A,nth(A,Xs),K2)) ) ) ) ).

% sum_list_update
tff(fact_4648_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image2(nat,nat,suc,top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_4649_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_mn(list(B),fun(A,list(product_prod(A,B))),Ys),Xs))) ).

% product_code
tff(fact_4650_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [F3: fun(nat,B),Ns: list(nat)] :
          ( ! [X3: nat,Y3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(nat,B,F3,X3)),aa(nat,B,F3,Y3))) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),aa(list(B),B,groups8242544230860333062m_list(B),map(nat,B,F3,Ns)))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_4651_range__mult,axiom,
    ! [A3: real] :
      ( ( ( A3 = zero_zero(real) )
       => ( image2(real,real,aa(real,fun(real,real),times_times(real),A3),top_top(set(real))) = aa(set(real),set(real),insert(real,zero_zero(real)),bot_bot(set(real))) ) )
      & ( ( A3 != zero_zero(real) )
       => ( image2(real,real,aa(real,fun(real,real),times_times(real),A3),top_top(set(real))) = top_top(set(real)) ) ) ) ).

% range_mult
tff(fact_4652_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P2: bool] :
      ( ( pp(P2)
       => ( aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_ne(bool,fun(A,fun(B,bool)),P2))) = top_top(set(product_prod(A,B))) ) )
      & ( ~ pp(P2)
       => ( aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_ne(bool,fun(A,fun(B,bool)),P2))) = bot_bot(set(product_prod(A,B))) ) ) ) ).

% Collect_const_case_prod
tff(fact_4653_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),top_top(fun(A,fun(B,bool))),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),top_top(set(product_prod(A,B))))) ) ).

% top_empty_eq2
tff(fact_4654_strict__sorted__imp__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% strict_sorted_imp_sorted
tff(fact_4655_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).

% sorted2
tff(fact_4656_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_4657_sorted__take,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),N: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),take(A,N,Xs)) ) ) ).

% sorted_take
tff(fact_4658_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [N: nat,X: A] : sorted_wrt(A,ord_less_eq(A),replicate(A,N,X)) ) ).

% sorted_replicate
tff(fact_4659_sorted__drop,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),N: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),drop(A,N,Xs)) ) ) ).

% sorted_drop
tff(fact_4660_sorted__tl,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),tl(A,Xs)) ) ) ).

% sorted_tl
tff(fact_4661_sorted__remdups__adj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remdups_adj(A,Xs)) ) ) ).

% sorted_remdups_adj
tff(fact_4662_sorted__upt,axiom,
    ! [M2: nat,N: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M2,N)) ).

% sorted_upt
tff(fact_4663_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] : sorted_wrt(A,ord_less_eq(A),linord4507533701916653071of_set(A,A6)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_4664_sorted__nths,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I5: set(nat)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),nths(A,Xs,I5)) ) ) ).

% sorted_nths
tff(fact_4665_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F3,Xs))
        <=> sorted_wrt(B,aTP_Lamp_nf(fun(B,A),fun(B,fun(B,bool)),F3),Xs) ) ) ).

% sorted_map
tff(fact_4666_surj__prod__decode,axiom,
    image2(nat,product_prod(nat,nat),nat_prod_decode,top_top(set(nat))) = top_top(set(product_prod(nat,nat))) ).

% surj_prod_decode
tff(fact_4667_bij__prod__decode,axiom,
    bij_betw(nat,product_prod(nat,nat),nat_prod_decode,top_top(set(nat)),top_top(set(product_prod(nat,nat)))) ).

% bij_prod_decode
tff(fact_4668_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ).

% sorted1
tff(fact_4669_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))
        <=> ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Ys)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) )
            & sorted_wrt(A,ord_less_eq(A),Ys) ) ) ) ).

% sorted_simps(2)
tff(fact_4670_surj__list__decode,axiom,
    image2(nat,list(nat),nat_list_decode,top_top(set(nat))) = top_top(set(list(nat))) ).

% surj_list_decode
tff(fact_4671_bij__list__decode,axiom,
    bij_betw(nat,list(nat),nat_list_decode,top_top(set(nat)),top_top(set(list(nat)))) ).

% bij_list_decode
tff(fact_4672_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_4673_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),append(A,Xs,Ys))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys)
            & ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
               => ! [Xa3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),aa(list(A),set(A),set2(A),Ys)))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa3)) ) ) ) ) ) ).

% sorted_append
tff(fact_4674_surj__list__encode,axiom,
    image2(list(nat),nat,nat_list_encode,top_top(set(list(nat)))) = top_top(set(nat)) ).

% surj_list_encode
tff(fact_4675_bij__list__encode,axiom,
    bij_betw(list(nat),nat,nat_list_encode,top_top(set(list(nat))),top_top(set(nat))) ).

% bij_list_encode
tff(fact_4676_surj__prod__encode,axiom,
    image2(product_prod(nat,nat),nat,nat_prod_encode,top_top(set(product_prod(nat,nat)))) = top_top(set(nat)) ).

% surj_prod_encode
tff(fact_4677_bij__prod__encode,axiom,
    bij_betw(product_prod(nat,nat),nat,nat_prod_encode,top_top(set(product_prod(nat,nat))),top_top(set(nat))) ).

% bij_prod_encode
tff(fact_4678_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),Xs: list(A),I2: nat,J2: nat] :
      ( sorted_wrt(A,P2,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),P2,aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J2))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_4679_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),Xs: list(A)] :
      ( sorted_wrt(A,P2,Xs)
    <=> ! [I: nat,J: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,aa(A,fun(A,bool),P2,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_4680_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( sorted_wrt(A,ord_less_eq(A),Ys)
             => ( distinct(A,Ys)
               => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
                 => ( Xs = Ys ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_4681_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,fun(A,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => sorted_wrt(A,P2,Xs) ) ).

% sorted_wrt01
tff(fact_4682_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I: nat,J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J))) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_4683_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted01
tff(fact_4684_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ? [X3: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X3) = A6 )
              & sorted_wrt(A,ord_less_eq(A),X3)
              & distinct(A,X3)
              & ! [Y4: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y4) = A6 )
                    & sorted_wrt(A,ord_less_eq(A),Y4)
                    & distinct(A,Y4) )
                 => ( Y4 = X3 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_4685_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( linord4507533701916653071of_set(A,aa(list(A),set(A),set2(A),Xs)) = Xs ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_4686_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I)))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_4687_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I2: nat,J2: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J2))) ) ) ) ) ).

% sorted_nth_mono
tff(fact_4688_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I: nat,J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J))) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_4689_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ~ ! [L3: list(A)] :
                ( sorted_wrt(A,ord_less(A),L3)
               => ( ( aa(list(A),set(A),set2(A),L3) = A6 )
                 => ( aa(list(A),nat,size_size(list(A)),L3) != aa(set(A),nat,finite_card(A),A6) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_4690_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I2: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Ns)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,nth(nat,Ns),I2))) ) ) ).

% sorted_wrt_less_idx
tff(fact_4691_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X5: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(list(A),set(A),set2(A),Xs)))
                & pp(aa(A,bool,P2,X5)) )
           => ( find(A,P2,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ng(list(A),fun(fun(A,bool),fun(A,bool)),Xs),P2)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_4692_sorted__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),map(product_prod(nat,A),nat,product_fst(nat,A),enumerate(A,N,Xs))) ).

% sorted_enumerate
tff(fact_4693_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),L: list(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A6 )
              & ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A6) ) )
          <=> ( linord4507533701916653071of_set(A,A6) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_4694_root__def,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(real,real,root(N),X) = zero_zero(real) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(real,real,root(N),X) = the_inv_into(real,real,top_top(set(real)),aTP_Lamp_nh(nat,fun(real,real),N),X) ) ) ) ).

% root_def
tff(fact_4695_length__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( ( ( Xs = nil(list(A)) )
         => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = zero_zero(nat) ) )
        & ( ( Xs != nil(list(A)) )
         => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat))) ) ) ) ) ).

% length_transpose_sorted
tff(fact_4696_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S3: set(B),F3: fun(B,A),A6: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),S3))
       => ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ~ ! [L3: list(B)] :
                ( sorted_wrt(A,Less,map(B,A,F3,L3))
               => ( ( aa(list(B),set(B),set2(B),L3) = A6 )
                 => ( aa(list(B),nat,size_size(list(B)),L3) != aa(set(B),nat,finite_card(B),A6) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_4697_length__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),rev(A,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rev
tff(fact_4698_top2I,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : pp(aa(B,bool,aa(A,fun(B,bool),top_top(fun(A,fun(B,bool))),X),Y)) ).

% top2I
tff(fact_4699_zip__rev,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( zip(A,B,rev(A,Xs),rev(B,Ys)) = rev(product_prod(A,B),zip(A,B,Xs,Ys)) ) ) ).

% zip_rev
tff(fact_4700_drop__rev,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,N,rev(A,Xs)) = rev(A,take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N),Xs)) ).

% drop_rev
tff(fact_4701_rev__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] : rev(A,drop(A,I2,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2),rev(A,Xs)) ).

% rev_drop
tff(fact_4702_rev__take,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] : rev(A,take(A,I2,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2),rev(A,Xs)) ).

% rev_take
tff(fact_4703_take__rev,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : take(A,N,rev(A,Xs)) = rev(A,drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N),Xs)) ).

% take_rev
tff(fact_4704_rotate__rev,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,N),rev(A,Xs)) = rev(A,aa(list(A),list(A),rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)))),Xs)) ).

% rotate_rev
tff(fact_4705_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_ni(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_4706_rev__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,rev(A,Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,suc,N))) ) ) ).

% rev_nth
tff(fact_4707_rev__update,axiom,
    ! [A: $tType,K2: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( rev(A,list_update(A,Xs,K2,Y)) = list_update(A,rev(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),K2)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_4708_sorted__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),transpose(A,Xs)))) ).

% sorted_transpose
tff(fact_4709_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I))),aa(nat,A,nth(A,Xs),I))) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_4710_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I: nat,J: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I))) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_4711_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I2: nat,J2: nat] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J2)),aa(nat,A,nth(A,Xs),I2))) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_4712_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat,J2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_nj(nat,fun(list(A),bool),I2),Xs))))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I2)),J2) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J2)),I2) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_4713_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( map(list(A),A,aTP_Lamp_nk(nat,fun(list(A),A),I2),filter2(list(A),aTP_Lamp_nj(nat,fun(list(A),bool),I2),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I2) ) ) ) ).

% transpose_column
tff(fact_4714_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S3: set(B),F3: fun(B,A),A6: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),S3))
       => ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ( sorted_wrt(A,Less,map(B,A,F3,L))
              & ( aa(list(B),set(B),set2(B),L) = A6 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A6) ) )
          <=> ( sorted8670434370408473282of_set(A,B,Less_eq,F3,A6) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_4715_length__concat__rev,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),nat,size_size(list(A)),concat(A,rev(list(A),Xs))) = aa(list(A),nat,size_size(list(A)),concat(A,Xs)) ).

% length_concat_rev
tff(fact_4716_length__filter__le,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_filter_le
tff(fact_4717_sum__length__filter__compl,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),filter2(A,aTP_Lamp_nl(fun(A,bool),fun(A,bool),P2),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_4718_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G3: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),filter2(A,aa(list(A),fun(A,bool),aTP_Lamp_nm(fun(list(A),A),fun(list(A),fun(A,bool)),G3),Xs),Xs)) ) ).

% sorted_same
tff(fact_4719_replicate__length__filter,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),filter2(A,aa(A,fun(A,bool),fequal(A),X),Xs)),X) = filter2(A,aa(A,fun(A,bool),fequal(A),X),Xs) ).

% replicate_length_filter
tff(fact_4720_length__filter__less,axiom,
    ! [A: $tType,X: A,Xs: list(A),P2: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(A,bool,P2,X))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% length_filter_less
tff(fact_4721_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P2: fun(B,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F3,Xs))
         => sorted_wrt(A,ord_less_eq(A),map(B,A,F3,filter2(B,P2,Xs))) ) ) ).

% sorted_filter
tff(fact_4722_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),G3: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),map(B,A,F3,filter2(B,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_nn(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),F3),G3),Xs),Xs))) ) ).

% sorted_map_same
tff(fact_4723_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F3: fun(B,A),P2: fun(B,bool),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F3,filter2(B,P2,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_no(fun(B,A),fun(fun(B,bool),fun(B,A)),F3),P2),Xs)) ) ).

% sum_list_map_filter'
tff(fact_4724_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F3: fun(A,nat),P2: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F3,filter2(A,P2,Xs)))),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F3,Xs)))) ).

% sum_list_filter_le_nat
tff(fact_4725_sum__list__map__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B),P2: fun(B,bool),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(list(B),set(B),set2(B),Xs)))
             => ( ~ pp(aa(B,bool,P2,X3))
               => ( aa(B,A,F3,X3) = zero_zero(A) ) ) )
         => ( aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F3,filter2(B,P2,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F3,Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_4726_filter__eq__nths,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : filter2(A,P2,Xs) = nths(A,Xs,aa(fun(nat,bool),set(nat),collect(nat),aa(list(A),fun(nat,bool),aTP_Lamp_np(fun(A,bool),fun(list(A),fun(nat,bool)),P2),Xs))) ).

% filter_eq_nths
tff(fact_4727_length__filter__conv__card,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(list(A),fun(nat,bool),aTP_Lamp_np(fun(A,bool),fun(list(A),fun(nat,bool)),P),Xs))) ).

% length_filter_conv_card
tff(fact_4728_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,bool)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs)) = aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,bool),set(A),collect(A),P2)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% distinct_length_filter
tff(fact_4729_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S3: set(B),F3: fun(B,A),A6: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),S3))
       => ( aa(list(B),nat,size_size(list(B)),sorted8670434370408473282of_set(A,B,Less_eq,F3,A6)) = aa(set(B),nat,finite_card(B),A6) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_4730_nth__transpose,axiom,
    ! [A: $tType,I2: nat,Xs: list(list(A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I2) = map(list(A),A,aTP_Lamp_nk(nat,fun(list(A),A),I2),filter2(list(A),aTP_Lamp_nj(nat,fun(list(A),bool),I2),Xs)) ) ) ).

% nth_transpose
tff(fact_4731_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_nj(nat,fun(list(A),bool),I2),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) ) ) ) ).

% transpose_column_length
tff(fact_4732_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : foldr(list(A),nat,aTP_Lamp_nq(list(A),fun(nat,nat)),transpose(A,Xs),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_nr(list(A),bool),Xs)) ).

% transpose_max_length
tff(fact_4733_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P2: fun(B,bool),Xs: list(B)] : map(B,A,F3,filter2(B,P2,Xs)) = map_filter(B,A,aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_ns(fun(B,A),fun(fun(B,bool),fun(B,option(A))),F3),P2),Xs) ).

% map_filter_map_filter
tff(fact_4734_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),foldr(list(B),nat,aTP_Lamp_nt(list(B),fun(nat,nat)),Xss,zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),foldr(list(B),nat,aTP_Lamp_nu(list(B),fun(nat,nat)),filter2(list(B),aTP_Lamp_nv(list(B),bool),Xss),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_4735_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),Xs) = foldr(A,A,plus_plus(A),Xs,zero_zero(A)) ) ).

% sum_list.eq_foldr
tff(fact_4736_nths__shift__lemma__Suc,axiom,
    ! [A: $tType,P2: fun(nat,bool),Xs: list(A),Is: list(nat)] : map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aTP_Lamp_nw(fun(nat,bool),fun(product_prod(A,nat),bool),P2),zip(A,nat,Xs,Is))) = map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aTP_Lamp_nx(fun(nat,bool),fun(product_prod(A,nat),bool),P2),zip(A,nat,Xs,map(nat,nat,suc,Is)))) ).

% nths_shift_lemma_Suc
tff(fact_4737_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F3),A3),Xs) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_ny(fun(B,A),fun(A,fun(B,fun(A,A))),F3),A3),Xs,zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_4738_nths__shift__lemma,axiom,
    ! [A: $tType,A6: set(nat),Xs: list(A),I2: nat] : map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aTP_Lamp_nz(set(nat),fun(product_prod(A,nat),bool),A6),zip(A,nat,Xs,upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))))) = map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_oa(set(nat),fun(nat,fun(product_prod(A,nat),bool)),A6),I2),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_4739_nths__def,axiom,
    ! [A: $tType,Xs: list(A),A6: set(nat)] : nths(A,Xs,A6) = map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aTP_Lamp_nz(set(nat),fun(product_prod(A,nat),bool),A6),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_4740_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = foldr(list(A),nat,aTP_Lamp_nq(list(A),fun(nat,nat)),Xs,zero_zero(nat)) ).

% length_transpose
tff(fact_4741_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( ( ( Xs = nil(A) )
             => ( foldr(A,A,ord_max(A),Xs,Y) = Y ) )
            & ( ( Xs != nil(A) )
             => ( foldr(A,A,ord_max(A),Xs,Y) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y) ) ) ) ) ) ).

% foldr_max_sorted
tff(fact_4742_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_nr(list(A),bool),Xs) ) ) ).

% transpose_transpose
tff(fact_4743_length__product__lists,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(list(B)),nat,size_size(list(list(B))),product_lists(B,Xss)) = foldr(nat,nat,times_times(nat),map(list(B),nat,size_size(list(B)),Xss),one_one(nat)) ).

% length_product_lists
tff(fact_4744_map__of__eq__Some__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
      ( distinct(A,map(product_prod(A,B),A,product_fst(A,B),Xys))
     => ( ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) )
      <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))) ) ) ).

% map_of_eq_Some_iff
tff(fact_4745_map__of__zip__is__None,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X) = none(B) )
      <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% map_of_zip_is_None
tff(fact_4746_map__of__is__SomeI,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),X: A,Y: B] :
      ( distinct(A,map(product_prod(A,B),A,product_fst(A,B),Xys))
     => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)))
       => ( aa(A,option(B),map_of(A,B,Xys),X) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_is_SomeI
tff(fact_4747_Some__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,X: A] :
      ( distinct(A,map(product_prod(A,B),A,product_fst(A,B),Xys))
     => ( ( aa(B,option(B),some(B),Y) = aa(A,option(B),map_of(A,B,Xys),X) )
      <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))) ) ) ).

% Some_eq_map_of_iff
tff(fact_4748_length__takeWhile__le,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_takeWhile_le
tff(fact_4749_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),takeWhile(A,P2,Xs)) ) ) ).

% sorted_takeWhile
tff(fact_4750_map__of__filter__in,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K2: B,Z2: A,P2: fun(B,fun(A,bool))] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K2) = aa(A,option(A),some(A),Z2) )
     => ( pp(aa(A,bool,aa(B,fun(A,bool),P2,K2),Z2))
       => ( aa(B,option(A),map_of(B,A,filter2(product_prod(B,A),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P2),Xs)),K2) = aa(A,option(A),some(A),Z2) ) ) ) ).

% map_of_filter_in
tff(fact_4751_takeWhile__eq__take,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : takeWhile(A,P2,Xs) = take(A,aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)),Xs) ).

% takeWhile_eq_take
tff(fact_4752_takeWhile__nth,axiom,
    ! [A: $tType,J2: nat,P2: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))))
     => ( aa(nat,A,nth(A,takeWhile(A,P2,Xs)),J2) = aa(nat,A,nth(A,Xs),J2) ) ) ).

% takeWhile_nth
tff(fact_4753_nth__length__takeWhile,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs)))
     => ~ pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))))) ) ).

% nth_length_takeWhile
tff(fact_4754_map__of__SomeD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K2: B,Y: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K2) = aa(A,option(A),some(A),Y) )
     => pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K2),Y)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs))) ) ).

% map_of_SomeD
tff(fact_4755_weak__map__of__SomeI,axiom,
    ! [A: $tType,B: $tType,K2: A,X: B,L: list(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),X)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L)))
     => ? [X3: B] : aa(A,option(B),map_of(A,B,L),K2) = aa(B,option(B),some(B),X3) ) ).

% weak_map_of_SomeI
tff(fact_4756_map__of__zip__inject,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xs: list(B),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Ys) = aa(list(B),nat,size_size(list(B)),Xs) )
     => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(B),nat,size_size(list(B)),Xs) )
       => ( distinct(B,Xs)
         => ( ( map_of(B,A,zip(B,A,Xs,Ys)) = map_of(B,A,zip(B,A,Xs,Zs)) )
           => ( Ys = Zs ) ) ) ) ) ).

% map_of_zip_inject
tff(fact_4757_map__of__Cons__code_I2_J,axiom,
    ! [C: $tType,B: $tType,L: B,K2: B,V3: C,Ps: list(product_prod(B,C))] :
      ( ( ( L = K2 )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(product_prod(B,C),fun(list(product_prod(B,C)),list(product_prod(B,C))),cons(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),L),V3)),Ps)),K2) = aa(C,option(C),some(C),V3) ) )
      & ( ( L != K2 )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(product_prod(B,C),fun(list(product_prod(B,C)),list(product_prod(B,C))),cons(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),L),V3)),Ps)),K2) = aa(B,option(C),map_of(B,C,Ps),K2) ) ) ) ).

% map_of_Cons_code(2)
tff(fact_4758_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J2: nat,P2: fun(A,bool),Xs: list(A)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J2))
         => pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),I3))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_4759_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A),P2: fun(A,bool)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),I3))) ) )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
         => ~ pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),N))) )
       => ( takeWhile(A,P2,Xs) = take(A,N,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_4760_map__of__zip__is__Some,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
      <=> ? [Y5: B] : aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X) = aa(B,option(B),some(B),Y5) ) ) ).

% map_of_zip_is_Some
tff(fact_4761_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F3: fun(A,B),X5: A] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,map(A,B,F3,Xs))),X5) = aa(B,option(B),some(B),aa(A,B,F3,X5)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,map(A,B,F3,Xs))),X5) = none(B) ) ) ) ).

% map_of_zip_map
tff(fact_4762_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),I2: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( distinct(A,Xs)
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Ys)))
         => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),aa(nat,A,nth(A,Xs),I2)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),I2)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_4763_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),T2: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,map(B,A,F3,Xs)))
         => ( filter2(B,aa(A,fun(B,bool),aTP_Lamp_ob(fun(B,A),fun(A,fun(B,bool)),F3),T2),Xs) = takeWhile(B,aa(A,fun(B,bool),aTP_Lamp_ob(fun(B,A),fun(A,fun(B,bool)),F3),T2),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_4764_set__map__of__compr,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
      ( distinct(A,map(product_prod(A,B),A,product_fst(A,B),Xs))
     => ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_oc(list(product_prod(A,B)),fun(A,fun(B,bool)),Xs))) ) ) ).

% set_map_of_compr
tff(fact_4765_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
         => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_4766_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D5: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( X != zero_zero(real) )
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
             => ( D5 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
               => ( D5 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
               => ( D5 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(N),D5,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_4767_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => has_field_derivative(real,aTP_Lamp_od(real,real),suminf(real,aTP_Lamp_oe(real,fun(nat,real),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_4768_has__field__derivative__sinh,axiom,
    ! [A9: $tType] :
      ( ( real_Vector_banach(A9)
        & real_V3459762299906320749_field(A9) )
     => ! [G3: fun(A9,A9),Db: A9,X: A9,S2: set(A9)] :
          ( has_field_derivative(A9,G3,Db,topolo174197925503356063within(A9,X,S2))
         => has_field_derivative(A9,aTP_Lamp_of(fun(A9,A9),fun(A9,A9),G3),aa(A9,A9,aa(A9,fun(A9,A9),times_times(A9),cosh(A9,aa(A9,A9,G3,X))),Db),topolo174197925503356063within(A9,X,S2)) ) ) ).

% has_field_derivative_sinh
tff(fact_4769_has__field__derivative__cosh,axiom,
    ! [A9: $tType] :
      ( ( real_Vector_banach(A9)
        & real_V3459762299906320749_field(A9) )
     => ! [G3: fun(A9,A9),Db: A9,X: A9,S2: set(A9)] :
          ( has_field_derivative(A9,G3,Db,topolo174197925503356063within(A9,X,S2))
         => has_field_derivative(A9,aTP_Lamp_og(fun(A9,A9),fun(A9,A9),G3),aa(A9,A9,aa(A9,fun(A9,A9),times_times(A9),sinh(A9,aa(A9,A9,G3,X))),Db),topolo174197925503356063within(A9,X,S2)) ) ) ).

% has_field_derivative_cosh
tff(fact_4770_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),M2: A,X: A] :
          ( has_field_derivative(A,G3,M2,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_oh(fun(A,A),fun(A,A),G3),aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,aa(A,A,G3,X))),M2),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_4771_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),M2: A,X: A] :
          ( has_field_derivative(A,G3,M2,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_oi(fun(A,A),fun(A,A),G3),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,aa(A,A,G3,X))),M2),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_sin
tff(fact_4772_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),M2: A,X: A] :
          ( has_field_derivative(A,G3,M2,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_oj(fun(A,A),fun(A,A),G3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,G3,X)))),M2),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_4773_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),X: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),C3),Y3))
         => has_field_derivative(A,aTP_Lamp_ok(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_hg(fun(nat,A),fun(A,fun(nat,A)),C3),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_4774_DERIV__fun__pow,axiom,
    ! [G3: fun(real,real),M2: real,X: real,N: nat] :
      ( has_field_derivative(real,G3,M2,topolo174197925503356063within(real,X,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_ol(fun(real,real),fun(nat,fun(real,real)),G3),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,G3,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))),M2),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_4775_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C3: fun(nat,A),F3: fun(A,A),F6: A,Z2: A] :
          ( ! [Z: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5))
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),C3),Z),aa(A,A,F3,Z)) )
         => ( has_field_derivative(A,F3,F6,topolo174197925503356063within(A,Z2,top_top(set(A))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K5))
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_hg(fun(nat,A),fun(A,fun(nat,A)),C3),Z2),F6) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_4776_has__real__derivative__powr,axiom,
    ! [Z2: real,R: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Z2))
     => has_field_derivative(real,aTP_Lamp_om(real,fun(real,real),R),aa(real,real,aa(real,fun(real,real),times_times(real),R),powr(real,Z2,aa(real,real,aa(real,fun(real,real),minus_minus(real),R),one_one(real)))),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_4777_DERIV__series_H,axiom,
    ! [F3: fun(real,fun(nat,real)),F6: fun(real,fun(nat,real)),X0: real,A3: real,B2: real,L5: fun(nat,real)] :
      ( ! [N3: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_on(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F3),N3),aa(nat,real,aa(real,fun(nat,real),F6,X0),N3),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,A3,B2)))
           => summable(real,aa(real,fun(nat,real),F3,X3)) )
       => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X0),set_or5935395276787703475ssThan(real,A3,B2)))
         => ( summable(real,aa(real,fun(nat,real),F6,X0))
           => ( summable(real,L5)
             => ( ! [N3: nat,X3: real,Y3: real] :
                    ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,A3,B2)))
                   => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Y3),set_or5935395276787703475ssThan(real,A3,B2)))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),F3,X3),N3)),aa(nat,real,aa(real,fun(nat,real),F3,Y3),N3)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,L5,N3)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X3),Y3))))) ) )
               => has_field_derivative(real,aTP_Lamp_oo(fun(real,fun(nat,real)),fun(real,real),F3),suminf(real,aa(real,fun(nat,real),F6,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_4778_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C3: fun(nat,A),Z2: A] :
          ( ! [Z: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5))
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),C3),Z)) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K5))
           => has_field_derivative(A,aTP_Lamp_ok(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_hg(fun(nat,A),fun(A,fun(nat,A)),C3),Z2)),topolo174197925503356063within(A,Z2,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_4779_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K5: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K5)))
           => has_field_derivative(A,aTP_Lamp_ok(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_hg(fun(nat,A),fun(A,fun(nat,A)),C3),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_4780_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K5: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hg(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_op(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K5)))
               => has_field_derivative(A,aTP_Lamp_ok(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_hg(fun(nat,A),fun(A,fun(nat,A)),C3),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_4781_DERIV__log,axiom,
    ! [X: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,log2(B2),divide_divide(real,one_one(real),aa(real,real,aa(real,fun(real,real),times_times(real),ln_ln(real,B2)),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_4782_DERIV__fun__powr,axiom,
    ! [G3: fun(real,real),M2: real,X: real,R: real] :
      ( has_field_derivative(real,G3,M2,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,G3,X)))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_oq(fun(real,real),fun(real,fun(real,real)),G3),R),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R),powr(real,aa(real,real,G3,X),aa(real,real,aa(real,fun(real,real),minus_minus(real),R),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M2),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_4783_DERIV__powr,axiom,
    ! [G3: fun(real,real),M2: real,X: real,F3: fun(real,real),R: real] :
      ( has_field_derivative(real,G3,M2,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,G3,X)))
       => ( has_field_derivative(real,F3,R,topolo174197925503356063within(real,X,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_or(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F3),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G3,X),aa(real,real,F3,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R),ln_ln(real,aa(real,real,G3,X)))),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),M2),aa(real,real,F3,X)),aa(real,real,G3,X)))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_4784_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_4785_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_4786_has__field__derivative__tanh,axiom,
    ! [A9: $tType] :
      ( ( real_Vector_banach(A9)
        & real_V3459762299906320749_field(A9) )
     => ! [G3: fun(A9,A9),X: A9,Db: A9,S2: set(A9)] :
          ( ( cosh(A9,aa(A9,A9,G3,X)) != zero_zero(A9) )
         => ( has_field_derivative(A9,G3,Db,topolo174197925503356063within(A9,X,S2))
           => has_field_derivative(A9,aTP_Lamp_os(fun(A9,A9),fun(A9,A9),G3),aa(A9,A9,aa(A9,fun(A9,A9),times_times(A9),aa(A9,A9,aa(A9,fun(A9,A9),minus_minus(A9),one_one(A9)),aa(nat,A9,aa(A9,fun(nat,A9),power_power(A9),aa(A9,A9,tanh(A9),aa(A9,A9,G3,X))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A9,X,S2)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_4787_DERIV__power__series_H,axiom,
    ! [R2: real,F3: fun(nat,real),X0: real] :
      ( ! [X3: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R2),R2)))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_ot(fun(nat,real),fun(real,fun(nat,real)),F3),X3)) )
     => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X0),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R2),R2)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
         => has_field_derivative(real,aTP_Lamp_ov(fun(nat,real),fun(real,real),F3),suminf(real,aa(real,fun(nat,real),aTP_Lamp_ot(fun(nat,real),fun(real,fun(nat,real)),F3),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_4788_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_4789_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),X: real,N: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( ! [M: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),X3),topolo174197925503356063within(real,X3,top_top(set(real))))
       => ? [T6: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),X)))
            & ( aa(real,real,F3,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_ow(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T6),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_4790_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),X: real,N: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
        & ! [M: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),X3),topolo174197925503356063within(real,X3,top_top(set(real)))) )
     => ? [T6: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),X)))
          & ( aa(real,real,F3,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_ow(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T6),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_4791_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( ( X != zero_zero(real) )
       => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_4792_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H),zero_zero(real)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
         => ( ! [M: nat,T6: real] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),H),T6))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T6),zero_zero(real))) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
           => ? [T6: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H),T6))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T6),zero_zero(real)))
                & ( aa(real,real,F3,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_ox(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T6),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_4793_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F3: fun(real,real),N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M: nat,T6: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T6))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T6),H)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
         => ? [T6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T6))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T6),H))
              & ( aa(real,real,F3,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_ox(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T6),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_4794_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
         => ( ! [M: nat,T6: real] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T6))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T6),H)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
           => ? [T6: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T6))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T6),H))
                & ( aa(real,real,F3,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_ox(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T6),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_4795_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),N: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( X != zero_zero(real) )
         => ( ! [M: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),X3),topolo174197925503356063within(real,X3,top_top(set(real))))
           => ? [T6: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T6)))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),X)))
                & ( aa(real,real,F3,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_ow(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T6),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_4796_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),N: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( ! [M: nat,T6: real] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),X))) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
       => ? [T6: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),X)))
            & ( aa(real,real,F3,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_ow(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T6),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_4797_Taylor__down,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A3: real,B2: real,C3: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M: nat,T6: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),T6))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T6),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),C3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C3),B2))
             => ? [T6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),T6))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T6),C3))
                  & ( aa(real,real,F3,A3) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_oy(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A3),C3)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T6),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),A3),C3)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_4798_Taylor__up,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A3: real,B2: real,C3: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M: nat,T6: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),T6))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T6),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),C3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C3),B2))
             => ? [T6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C3),T6))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T6),B2))
                  & ( aa(real,real,F3,B2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_oy(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C3)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T6),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),C3)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_4799_Taylor,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A3: real,B2: real,C3: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M: nat,T6: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),T6))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T6),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),C3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C3),B2))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),B2))
                 => ( ( X != C3 )
                   => ? [T6: real] :
                        ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),C3))
                         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),T6))
                            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T6),C3)) ) )
                        & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),C3))
                         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C3),T6))
                            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T6),X)) ) )
                        & ( aa(real,real,F3,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_oz(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C3),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T6),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),C3)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_4800_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun(nat,fun(real,real)),K2: nat,B6: real] :
      ( ! [M: nat,T6: real] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T6))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T6),H)) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
     => ( ( N = aa(nat,nat,suc,K2) )
       => ! [M3: nat,T7: real] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T7))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T7),H)) )
           => has_field_derivative(real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_pb(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),N),Diff),B6),M3),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T7)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_pc(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M3),T7)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,M3))))),aa(real,real,aa(real,fun(real,real),times_times(real),B6),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),T7),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,M3))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,M3))))))),topolo174197925503356063within(real,T7,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_4801_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D3: A,X: A,S2: set(A)] :
          ( has_field_derivative(A,F3,D3,topolo174197925503356063within(A,X,S2))
         => ( ( aa(A,A,F3,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_pd(fun(A,A),fun(A,A),F3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F3,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_4802_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D3: A,X: A,S2: set(A),G3: fun(A,A),E3: A] :
          ( has_field_derivative(A,F3,D3,topolo174197925503356063within(A,X,S2))
         => ( has_field_derivative(A,G3,E3,topolo174197925503356063within(A,X,S2))
           => ( ( aa(A,A,G3,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pe(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,G3,X))),aa(A,A,aa(A,fun(A,A),times_times(A),E3),aa(A,A,F3,X))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,G3,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% DERIV_quotient
tff(fact_4803_DERIV__pow,axiom,
    ! [N: nat,X: real,S2: set(real)] : has_field_derivative(real,aTP_Lamp_pf(nat,fun(real,real),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,X,S2)) ).

% DERIV_pow
tff(fact_4804_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C3: A] : aTP_Lamp_pg(A,fun(A,A),C3) = aa(A,fun(A,A),times_times(A),C3) ) ).

% mult_commute_abs
tff(fact_4805_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K2: A,F4: filter(A)] : has_field_derivative(A,aTP_Lamp_ph(A,fun(A,A),K2),zero_zero(A),F4) ) ).

% DERIV_const
tff(fact_4806_DERIV__cmult__Id,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,X: A,S2: set(A)] : has_field_derivative(A,aa(A,fun(A,A),times_times(A),C3),C3,topolo174197925503356063within(A,X,S2)) ) ).

% DERIV_cmult_Id
tff(fact_4807_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,X: A,S2: set(A),C3: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,X,S2))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_pi(fun(A,A),fun(A,fun(A,A)),F3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),D5),C3),topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_cmult_right
tff(fact_4808_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,X: A,S2: set(A),C3: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,X,S2))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_pj(fun(A,A),fun(A,fun(A,A)),F3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D5),topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_cmult
tff(fact_4809_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,X: A,S2: set(A),G3: fun(A,A),E5: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,X,S2))
         => ( has_field_derivative(A,G3,E5,topolo174197925503356063within(A,X,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pk(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F3,X)),E5)),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(A,A,G3,X))),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_mult'
tff(fact_4810_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Da: A,X: A,S2: set(A),G3: fun(A,A),Db: A] :
          ( has_field_derivative(A,F3,Da,topolo174197925503356063within(A,X,S2))
         => ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,X,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pk(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Da),aa(A,A,G3,X))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F3,X))),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_mult
tff(fact_4811_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,X: A,S2: set(A),G3: fun(A,A),E5: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,X,S2))
         => ( has_field_derivative(A,G3,E5,topolo174197925503356063within(A,aa(A,A,F3,X),top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pl(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),times_times(A),E5),D5),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_chain'
tff(fact_4812_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Da: A,G3: fun(A,A),X: A,Db: A,S2: set(A)] :
          ( has_field_derivative(A,F3,Da,topolo174197925503356063within(A,aa(A,A,G3,X),top_top(set(A))))
         => ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,X,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pm(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_chain2
tff(fact_4813_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G3: fun(A,A),G5: fun(A,A),F3: fun(A,A),F6: A,X: A] :
          ( ! [X3: A] : has_field_derivative(A,G3,aa(A,A,G5,X3),topolo174197925503356063within(A,X3,top_top(set(A))))
         => ( has_field_derivative(A,F3,F6,topolo174197925503356063within(A,X,top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pm(fun(A,A),fun(fun(A,A),fun(A,A)),G3),F3),aa(A,A,aa(A,fun(A,A),times_times(A),F6),aa(A,A,G5,aa(A,A,F3,X))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_4814_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [S2: set(A),G3: fun(A,A),G5: fun(A,A),F3: fun(A,A),F6: A,X: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
             => has_field_derivative(A,G3,aa(A,A,G5,X3),topolo174197925503356063within(A,X3,top_top(set(A)))) )
         => ( has_field_derivative(A,F3,F6,topolo174197925503356063within(A,X,top_top(set(A))))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,F3,X)),S2))
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pm(fun(A,A),fun(fun(A,A),fun(A,A)),G3),F3),aa(A,A,aa(A,fun(A,A),times_times(A),F6),aa(A,A,G5,aa(A,A,F3,X))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_4815_DERIV__const__ratio__const,axiom,
    ! [A3: real,B2: real,F3: fun(real,real),K2: real] :
      ( ( A3 != B2 )
     => ( ! [X3: real] : has_field_derivative(real,F3,K2,topolo174197925503356063within(real,X3,top_top(set(real))))
       => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3)),K2) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_4816_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,X: A,S2: set(A),G3: fun(A,A),E5: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,X,S2))
         => ( has_field_derivative(A,G3,E5,topolo174197925503356063within(A,X,S2))
           => ( ( aa(A,A,G3,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_pe(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(A,A,G3,X))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F3,X)),E5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G3,X)),aa(A,A,G3,X))),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% DERIV_divide
tff(fact_4817_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,X: A,S2: set(A)] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,X,S2))
         => ( ( aa(A,A,F3,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_pd(fun(A,A),fun(A,A),F3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(A,A,F3,X))),D5)),aa(A,A,inverse_inverse(A),aa(A,A,F3,X)))),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_inverse'
tff(fact_4818_MVT2,axiom,
    ! [A3: real,B2: real,F3: fun(real,real),F6: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
             => has_field_derivative(real,F3,aa(real,real,F6,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
       => ? [Z: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),Z))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z),B2))
            & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3)),aa(real,real,F6,Z)) ) ) ) ) ).

% MVT2
tff(fact_4819_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,X: A,S2: set(A),N: nat] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,X,S2))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_pn(fun(A,A),fun(nat,fun(A,A)),F3),N),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),N))),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F3,X)),N))),topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_power_Suc
tff(fact_4820_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,S2: set(A)] :
          ( ( X != zero_zero(A) )
         => has_field_derivative(A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_inverse
tff(fact_4821_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,X: A,S2: set(A),N: nat] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,X,S2))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_po(fun(A,A),fun(nat,fun(A,A)),F3),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F3,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_power
tff(fact_4822_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),X: A,G5: fun(A,real),S2: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G3,X)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,G3,X)),one_one(real)))
           => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,X,S2))
             => has_derivative(A,real,aTP_Lamp_pp(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pq(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),X),G5),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_4823_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),X: A,G5: fun(A,real),S2: set(A)] :
          ( ( cos(real,aa(A,real,G3,X)) != zero_zero(real) )
         => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,X,S2))
           => has_derivative(A,real,aTP_Lamp_pr(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_ps(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),X),G5),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivative_tan
tff(fact_4824_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),G5: fun(A,real),X: A,S2: set(A)] :
          ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,X,S2))
         => has_derivative(A,real,aTP_Lamp_pt(fun(A,real),fun(A,real),G3),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_pu(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G3),G5),X),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_arctan
tff(fact_4825_has__derivative__mult__right,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V4412858255891104859lgebra(A) )
     => ! [G3: fun(C,A),G5: fun(C,A),F4: filter(C),X: A] :
          ( has_derivative(C,A,G3,G5,F4)
         => has_derivative(C,A,aa(A,fun(C,A),aTP_Lamp_pv(fun(C,A),fun(A,fun(C,A)),G3),X),aa(A,fun(C,A),aTP_Lamp_pv(fun(C,A),fun(A,fun(C,A)),G5),X),F4) ) ) ).

% has_derivative_mult_right
tff(fact_4826_has__derivative__mult__left,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V4412858255891104859lgebra(A) )
     => ! [G3: fun(C,A),G5: fun(C,A),F4: filter(C),Y: A] :
          ( has_derivative(C,A,G3,G5,F4)
         => has_derivative(C,A,aa(A,fun(C,A),aTP_Lamp_pw(fun(C,A),fun(A,fun(C,A)),G3),Y),aa(A,fun(C,A),aTP_Lamp_pw(fun(C,A),fun(A,fun(C,A)),G5),Y),F4) ) ) ).

% has_derivative_mult_left
tff(fact_4827_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C3: B,F4: filter(A)] : has_derivative(A,B,aTP_Lamp_px(B,fun(A,B),C3),aTP_Lamp_py(A,B),F4) ) ).

% has_derivative_const
tff(fact_4828_has__field__derivative__imp__has__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,F4: filter(A)] :
          ( has_field_derivative(A,F3,D5,F4)
         => has_derivative(A,A,F3,aa(A,fun(A,A),times_times(A),D5),F4) ) ) ).

% has_field_derivative_imp_has_derivative
tff(fact_4829_has__derivative__imp__has__field__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: fun(A,A),F4: filter(A),D7: A] :
          ( has_derivative(A,A,F3,D5,F4)
         => ( ! [X3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X3),D7) = aa(A,A,D5,X3)
           => has_field_derivative(A,F3,D7,F4) ) ) ) ).

% has_derivative_imp_has_field_derivative
tff(fact_4830_has__field__derivative__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,F4: filter(A)] :
          ( has_field_derivative(A,F3,D5,F4)
        <=> has_derivative(A,A,F3,aa(A,fun(A,A),times_times(A),D5),F4) ) ) ).

% has_field_derivative_def
tff(fact_4831_has__derivative__mult,axiom,
    ! [A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [F3: fun(D,A),F6: fun(D,A),X: D,S2: set(D),G3: fun(D,A),G5: fun(D,A)] :
          ( has_derivative(D,A,F3,F6,topolo174197925503356063within(D,X,S2))
         => ( has_derivative(D,A,G3,G5,topolo174197925503356063within(D,X,S2))
           => has_derivative(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_pz(fun(D,A),fun(fun(D,A),fun(D,A)),F3),G3),aa(fun(D,A),fun(D,A),aa(fun(D,A),fun(fun(D,A),fun(D,A)),aa(D,fun(fun(D,A),fun(fun(D,A),fun(D,A))),aa(fun(D,A),fun(D,fun(fun(D,A),fun(fun(D,A),fun(D,A)))),aTP_Lamp_qa(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,A),fun(fun(D,A),fun(D,A))))),F3),F6),X),G3),G5),topolo174197925503356063within(D,X,S2)) ) ) ) ).

% has_derivative_mult
tff(fact_4832_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F4: fun(A,B),X: A] :
          ( has_derivative(A,B,aTP_Lamp_py(A,B),F4,topolo174197925503356063within(A,X,top_top(set(A))))
         => ! [X5: A] : aa(A,B,F4,X5) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_4833_has__derivative__exp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),G5: fun(A,real),X: A,S2: set(A)] :
          ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,X,S2))
         => has_derivative(A,real,aTP_Lamp_qb(fun(A,real),fun(A,real),G3),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_qc(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G3),G5),X),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_exp
tff(fact_4834_has__derivative__sin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),G5: fun(A,real),X: A,S2: set(A)] :
          ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,X,S2))
         => has_derivative(A,real,aTP_Lamp_qd(fun(A,real),fun(A,real),G3),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_qe(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G3),G5),X),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_sin
tff(fact_4835_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),Db: A,X: A,S2: set(A)] :
          ( has_derivative(A,A,G3,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,S2))
         => has_derivative(A,A,aTP_Lamp_qf(fun(A,A),fun(A,A),G3),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G3,X))),Db)),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_sinh
tff(fact_4836_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),Db: A,X: A,S2: set(A)] :
          ( has_derivative(A,A,G3,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,S2))
         => has_derivative(A,A,aTP_Lamp_qg(fun(A,A),fun(A,A),G3),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G3,X))),Db)),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_cosh
tff(fact_4837_has__derivative__divide_H,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [F3: fun(C,A),F6: fun(C,A),X: C,S3: set(C),G3: fun(C,A),G5: fun(C,A)] :
          ( has_derivative(C,A,F3,F6,topolo174197925503356063within(C,X,S3))
         => ( has_derivative(C,A,G3,G5,topolo174197925503356063within(C,X,S3))
           => ( ( aa(C,A,G3,X) != zero_zero(A) )
             => has_derivative(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_qh(fun(C,A),fun(fun(C,A),fun(C,A)),F3),G3),aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_qi(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),F3),F6),X),G3),G5),topolo174197925503356063within(C,X,S3)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_4838_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,S3: set(A)] :
          ( ( X != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_qj(A,fun(A,A),X),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_inverse'
tff(fact_4839_has__derivative__inverse,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(C,A),X: C,F6: fun(C,A),S3: set(C)] :
          ( ( aa(C,A,F3,X) != zero_zero(A) )
         => ( has_derivative(C,A,F3,F6,topolo174197925503356063within(C,X,S3))
           => has_derivative(C,A,aTP_Lamp_qk(fun(C,A),fun(C,A),F3),aa(fun(C,A),fun(C,A),aa(C,fun(fun(C,A),fun(C,A)),aTP_Lamp_ql(fun(C,A),fun(C,fun(fun(C,A),fun(C,A))),F3),X),F6),topolo174197925503356063within(C,X,S3)) ) ) ) ).

% has_derivative_inverse
tff(fact_4840_DERIV__compose__FDERIV,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(real,real),F6: real,G3: fun(A,real),X: A,G5: fun(A,real),S2: set(A)] :
          ( has_field_derivative(real,F3,F6,topolo174197925503356063within(real,aa(A,real,G3,X),top_top(set(real))))
         => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,X,S2))
           => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qm(fun(real,real),fun(fun(A,real),fun(A,real)),F3),G3),aa(fun(A,real),fun(A,real),aTP_Lamp_qn(real,fun(fun(A,real),fun(A,real)),F6),G5),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_compose_FDERIV
tff(fact_4841_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),G5: fun(A,real),X: A,S2: set(A)] :
          ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,X,S2))
         => has_derivative(A,real,aTP_Lamp_qo(fun(A,real),fun(A,real),G3),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_qp(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G3),G5),X),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_cos
tff(fact_4842_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),X: A,S3: set(A),N: nat] :
          ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,X,S3))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_qq(fun(A,B),fun(nat,fun(A,B)),F3),N),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_qr(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F3),F6),X),N),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_power
tff(fact_4843_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),X: A,G5: fun(A,real),S2: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G3,X)))
         => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,X,S2))
           => has_derivative(A,real,aTP_Lamp_qs(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_qt(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),X),G5),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivative_ln
tff(fact_4844_has__derivative__divide,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(C,A),F6: fun(C,A),X: C,S3: set(C),G3: fun(C,A),G5: fun(C,A)] :
          ( has_derivative(C,A,F3,F6,topolo174197925503356063within(C,X,S3))
         => ( has_derivative(C,A,G3,G5,topolo174197925503356063within(C,X,S3))
           => ( ( aa(C,A,G3,X) != zero_zero(A) )
             => has_derivative(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_qu(fun(C,A),fun(fun(C,A),fun(C,A)),F3),G3),aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_qv(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),F3),F6),X),G3),G5),topolo174197925503356063within(C,X,S3)) ) ) ) ) ).

% has_derivative_divide
tff(fact_4845_has__derivative__prod,axiom,
    ! [B: $tType,I6: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [I5: set(I6),F3: fun(I6,fun(A,B)),F6: fun(I6,fun(A,B)),X: A,S3: set(A)] :
          ( ! [I3: I6] :
              ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I3),I5))
             => has_derivative(A,B,aa(I6,fun(A,B),F3,I3),aa(I6,fun(A,B),F6,I3),topolo174197925503356063within(A,X,S3)) )
         => has_derivative(A,B,aa(fun(I6,fun(A,B)),fun(A,B),aTP_Lamp_qx(set(I6),fun(fun(I6,fun(A,B)),fun(A,B)),I5),F3),aa(A,fun(A,B),aa(fun(I6,fun(A,B)),fun(A,fun(A,B)),aa(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B))),aTP_Lamp_qz(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B)))),I5),F3),F6),X),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_prod
tff(fact_4846_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),G5: fun(A,real),X: A,X6: set(A),F3: fun(A,real),F6: fun(A,real)] :
          ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,X,X6))
         => ( has_derivative(A,real,F3,F6,topolo174197925503356063within(A,X,X6))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G3,X)))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ra(fun(A,real),fun(fun(A,real),fun(A,real)),G3),F3),aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_rb(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G3),G5),X),F3),F6),topolo174197925503356063within(A,X,X6)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_4847_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),X: A,G5: fun(A,real),S2: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G3,X)))
         => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,X,S2))
           => has_derivative(A,real,aTP_Lamp_rc(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_rd(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),X),G5),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_4848_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),X: A,G5: fun(A,real),S2: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G3,X)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,G3,X)),one_one(real)))
           => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,X,S2))
             => has_derivative(A,real,aTP_Lamp_re(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_rf(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),X),G5),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_4849_has__derivative__floor,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & archim2362893244070406136eiling(Aa)
        & topolo2564578578187576103pology(Aa) )
     => ! [G3: fun(A,real),X: A,F3: fun(real,Aa),G5: fun(A,real),S2: set(A)] :
          ( topolo3448309680560233919inuous(real,Aa,topolo174197925503356063within(real,aa(A,real,G3,X),top_top(set(real))),F3)
         => ( ~ pp(aa(set(Aa),bool,aa(Aa,fun(set(Aa),bool),member(Aa),aa(real,Aa,F3,aa(A,real,G3,X))),ring_1_Ints(Aa)))
           => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,X,S2))
             => has_derivative(A,real,aa(fun(real,Aa),fun(A,real),aTP_Lamp_rg(fun(A,real),fun(fun(real,Aa),fun(A,real)),G3),F3),aTP_Lamp_rh(fun(A,real),fun(A,real),G5),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% has_derivative_floor
tff(fact_4850_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K5: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_op(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K5)))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rj(fun(nat,A),fun(A,fun(A,A)),C3),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_4851_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F3: fun(B,A),L: A,F4: filter(B)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rk(A,fun(fun(B,A),fun(B,A)),C3),F3),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C3)),F4)
          <=> filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_4852_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F3: fun(B,A),L: A,F4: filter(B)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rl(A,fun(fun(B,A),fun(B,A)),C3),F3),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),L)),F4)
          <=> filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_4853_power__tendsto__0__iff,axiom,
    ! [A: $tType,N: nat,F3: fun(A,real),F4: filter(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rm(nat,fun(fun(A,real),fun(A,real)),N),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
      <=> filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% power_tendsto_0_iff
tff(fact_4854_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: fun(A,A),A3: A,F4: filter(A)] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( ( cos(A,A3) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_rn(fun(A,A),fun(A,A),F3),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A3)),F4) ) ) ) ).

% tendsto_tan
tff(fact_4855_tendsto__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [F3: fun(B,A),A3: A,F4: filter(B),G3: fun(B,A),B2: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( filterlim(B,A,G3,topolo7230453075368039082e_nhds(A,B2),F4)
           => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ro(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),F4) ) ) ) ).

% tendsto_mult
tff(fact_4856_tendsto__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [F3: fun(B,A),L: A,F4: filter(B),C3: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4)
         => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_rp(fun(B,A),fun(A,fun(B,A)),F3),C3),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),L)),F4) ) ) ).

% tendsto_mult_left
tff(fact_4857_tendsto__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [F3: fun(B,A),L: A,F4: filter(B),C3: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4)
         => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_rq(fun(B,A),fun(A,fun(B,A)),F3),C3),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C3)),F4) ) ) ).

% tendsto_mult_right
tff(fact_4858_tendsto__mult__one,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F3: fun(D,B),F4: filter(D),G3: fun(D,B)] :
          ( filterlim(D,B,F3,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
         => ( filterlim(D,B,G3,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
           => filterlim(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_rr(fun(D,B),fun(fun(D,B),fun(D,B)),F3),G3),topolo7230453075368039082e_nhds(B,one_one(B)),F4) ) ) ) ).

% tendsto_mult_one
tff(fact_4859_continuous__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topological_t2_space(B) )
     => ! [F4: filter(B),F3: fun(B,A),C3: A] :
          ( topolo3448309680560233919inuous(B,A,F4,F3)
         => topolo3448309680560233919inuous(B,A,F4,aa(A,fun(B,A),aTP_Lamp_rs(fun(B,A),fun(A,fun(B,A)),F3),C3)) ) ) ).

% continuous_mult_right
tff(fact_4860_continuous__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topological_t2_space(B) )
     => ! [F4: filter(B),F3: fun(B,A),C3: A] :
          ( topolo3448309680560233919inuous(B,A,F4,F3)
         => topolo3448309680560233919inuous(B,A,F4,aa(A,fun(B,A),aTP_Lamp_rt(fun(B,A),fun(A,fun(B,A)),F3),C3)) ) ) ).

% continuous_mult_left
tff(fact_4861_continuous__mult_H,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topological_t2_space(D)
        & topolo4211221413907600880p_mult(B) )
     => ! [F4: filter(D),F3: fun(D,B),G3: fun(D,B)] :
          ( topolo3448309680560233919inuous(D,B,F4,F3)
         => ( topolo3448309680560233919inuous(D,B,F4,G3)
           => topolo3448309680560233919inuous(D,B,F4,aa(fun(D,B),fun(D,B),aTP_Lamp_ru(fun(D,B),fun(fun(D,B),fun(D,B)),F3),G3)) ) ) ) ).

% continuous_mult'
tff(fact_4862_continuous__mult,axiom,
    ! [A: $tType,D: $tType] :
      ( ( topological_t2_space(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [F4: filter(D),F3: fun(D,A),G3: fun(D,A)] :
          ( topolo3448309680560233919inuous(D,A,F4,F3)
         => ( topolo3448309680560233919inuous(D,A,F4,G3)
           => topolo3448309680560233919inuous(D,A,F4,aa(fun(D,A),fun(D,A),aTP_Lamp_rv(fun(D,A),fun(fun(D,A),fun(D,A)),F3),G3)) ) ) ) ).

% continuous_mult
tff(fact_4863_tendsto__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(B,A),A3: A,F4: filter(B),G3: fun(B,A),B2: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( filterlim(B,A,G3,topolo7230453075368039082e_nhds(A,B2),F4)
           => ( ( B2 != zero_zero(A) )
             => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rw(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3),topolo7230453075368039082e_nhds(A,divide_divide(A,A3,B2)),F4) ) ) ) ) ).

% tendsto_divide
tff(fact_4864_tendsto__divide__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(B,A),F4: filter(B),C3: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_rx(fun(B,A),fun(A,fun(B,A)),F3),C3),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ).

% tendsto_divide_zero
tff(fact_4865_tendsto__mult__right__zero,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(D,A),F4: filter(D),C3: A] :
          ( filterlim(D,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => filterlim(D,A,aa(A,fun(D,A),aTP_Lamp_ry(fun(D,A),fun(A,fun(D,A)),F3),C3),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ).

% tendsto_mult_right_zero
tff(fact_4866_tendsto__mult__left__zero,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(D,A),F4: filter(D),C3: A] :
          ( filterlim(D,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => filterlim(D,A,aa(A,fun(D,A),aTP_Lamp_rz(fun(D,A),fun(A,fun(D,A)),F3),C3),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ).

% tendsto_mult_left_zero
tff(fact_4867_tendsto__mult__zero,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(D,A),F4: filter(D),G3: fun(D,A)] :
          ( filterlim(D,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => ( filterlim(D,A,G3,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_sa(fun(D,A),fun(fun(D,A),fun(D,A)),F3),G3),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ) ).

% tendsto_mult_zero
tff(fact_4868_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_sb(fun(A,B),fun(B,fun(A,B)),F3),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% LIM_zero
tff(fact_4869_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_sb(fun(A,B),fun(B,fun(A,B)),F3),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
        <=> filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_iff
tff(fact_4870_Lim__transform,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(B,A),A3: A,F4: filter(B),F3: fun(B,A)] :
          ( filterlim(B,A,G3,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_sc(fun(B,A),fun(fun(B,A),fun(B,A)),G3),F3),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,A3),F4) ) ) ) ).

% Lim_transform
tff(fact_4871_Lim__transform2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(B,A),A3: A,F4: filter(B),G3: fun(B,A)] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_sd(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(B,A,G3,topolo7230453075368039082e_nhds(A,A3),F4) ) ) ) ).

% Lim_transform2
tff(fact_4872_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_sb(fun(A,B),fun(B,fun(A,B)),F3),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_cancel
tff(fact_4873_Lim__transform__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(B,A),G3: fun(B,A),F4: filter(B),A3: A] :
          ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_sd(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
         => ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,A3),F4)
          <=> filterlim(B,A,G3,topolo7230453075368039082e_nhds(A,A3),F4) ) ) ) ).

% Lim_transform_eq
tff(fact_4874_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,real,aTP_Lamp_se(fun(A,B),fun(A,real),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_norm_zero
tff(fact_4875_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_se(fun(A,B),fun(A,real),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
        <=> filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_iff
tff(fact_4876_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_se(fun(A,B),fun(A,real),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_4877_tendsto__null__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(B),F3: fun(A,fun(B,C)),F4: filter(A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),I5))
             => filterlim(A,C,aa(B,fun(A,C),aTP_Lamp_sf(fun(A,fun(B,C)),fun(B,fun(A,C)),F3),I3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) )
         => filterlim(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_sg(set(B),fun(fun(A,fun(B,C)),fun(A,C)),I5),F3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ).

% tendsto_null_sum
tff(fact_4878_tendsto__sgn,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(B,A),L: A,F4: filter(B)] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4)
         => ( ( L != zero_zero(A) )
           => filterlim(B,A,aTP_Lamp_sh(fun(B,A),fun(B,A),F3),topolo7230453075368039082e_nhds(A,aa(A,A,sgn_sgn(A),L)),F4) ) ) ) ).

% tendsto_sgn
tff(fact_4879_continuous__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,C,F4,G3)
           => topolo3448309680560233919inuous(A,product_prod(B,C),F4,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_si(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3)) ) ) ) ).

% continuous_Pair
tff(fact_4880_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: fun(A,B),A3: B,F4: filter(A),G3: fun(A,C),B2: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A3),F4)
         => ( filterlim(A,C,G3,topolo7230453075368039082e_nhds(C,B2),F4)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_sj(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3),topolo7230453075368039082e_nhds(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)),F4) ) ) ) ).

% tendsto_Pair
tff(fact_4881_tendsto__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(B,A),A3: A,F4: filter(B)] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( ( A3 != zero_zero(A) )
           => filterlim(B,A,aTP_Lamp_sk(fun(B,A),fun(B,A),F3),topolo7230453075368039082e_nhds(A,aa(A,A,inverse_inverse(A),A3)),F4) ) ) ) ).

% tendsto_inverse
tff(fact_4882_tendsto__add__zero,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F3: fun(D,B),F4: filter(D),G3: fun(D,B)] :
          ( filterlim(D,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(D,B,G3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_sl(fun(D,B),fun(fun(D,B),fun(D,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_add_zero
tff(fact_4883_tendsto__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: fun(C,A),A3: A,F4: filter(C)] :
          ( filterlim(C,A,F3,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( ( cosh(A,A3) != zero_zero(A) )
           => filterlim(C,A,aTP_Lamp_sm(fun(C,A),fun(C,A),F3),topolo7230453075368039082e_nhds(A,aa(A,A,tanh(A),A3)),F4) ) ) ) ).

% tendsto_tanh
tff(fact_4884_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: fun(A,A),A3: A,F4: filter(A)] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( ( sin(A,A3) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_sn(fun(A,A),fun(A,A),F3),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A3)),F4) ) ) ) ).

% tendsto_cot
tff(fact_4885_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [X: A,F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),F3)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_so(A,fun(fun(A,B),fun(A,B)),X),F3),topolo7230453075368039082e_nhds(B,aa(A,B,F3,X)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_4886_isCont__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,F3: fun(A,B),G3: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A3,top_top(set(A))),G3)
           => topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_si(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3)) ) ) ) ).

% isCont_Pair
tff(fact_4887_LIM__not__zero,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( topolo8386298272705272623_space(A)
        & zero(Aa)
        & topological_t2_space(Aa) )
     => ! [K2: Aa,A3: A] :
          ( ( K2 != zero_zero(Aa) )
         => ~ filterlim(A,Aa,aTP_Lamp_sp(Aa,fun(A,Aa),K2),topolo7230453075368039082e_nhds(Aa,zero_zero(Aa)),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_not_zero
tff(fact_4888_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A3: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,A3)),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_sq(fun(A,B),fun(A,fun(A,B)),F3),A3),topolo7230453075368039082e_nhds(B,aa(A,B,F3,A3)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_4889_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),L5: B,A3: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_sq(fun(A,B),fun(A,fun(A,B)),F3),A3),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_4890_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A3: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_sq(fun(A,B),fun(A,fun(A,B)),F3),A3),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_4891_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F3: fun(A,B),F4: filter(A),N: nat] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_sr(fun(A,B),fun(nat,fun(A,B)),F3),N),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_null_power
tff(fact_4892_LIM__offset__zero__iff,axiom,
    ! [C: $tType,D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D)
        & zero(C) )
     => ! [A3: A,F3: fun(A,D),L5: D] :
          ( nO_MATCH(C,A,zero_zero(C),A3)
         => ( filterlim(A,D,F3,topolo7230453075368039082e_nhds(D,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
          <=> filterlim(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_ss(A,fun(fun(A,D),fun(A,D)),A3),F3),topolo7230453075368039082e_nhds(D,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_4893_IVT2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F3: fun(A,B),B2: A,Y: B,A3: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,B2)),Y))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),aa(A,B,F3,A3)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
             => ( ! [X3: A] :
                    ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2)) )
                   => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X3,top_top(set(A))),F3) )
               => ? [X3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
                    & ( aa(A,B,F3,X3) = Y ) ) ) ) ) ) ) ).

% IVT2
tff(fact_4894_IVT,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F3: fun(A,B),A3: A,Y: B,B2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,A3)),Y))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),aa(A,B,F3,B2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
             => ( ! [X3: A] :
                    ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2)) )
                   => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X3,top_top(set(A))),F3) )
               => ? [X3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
                    & ( aa(A,B,F3,X3) = Y ) ) ) ) ) ) ) ).

% IVT
tff(fact_4895_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(A,A),A3: A,D5: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_st(fun(A,A),fun(A,fun(A,A)),F3),A3),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_su(fun(A,A),fun(A,fun(A,A)),F3),A3),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_4896_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,S2: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S2),F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S2),G3)
           => ( ( aa(A,B,G3,A3) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S2),aa(fun(A,B),fun(A,B),aTP_Lamp_sv(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_4897_isCont__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [A3: A,F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G3)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_sw(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% isCont_mult
tff(fact_4898_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A3: A,S2: set(A),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S2),F3)
         => ( ( aa(A,B,F3,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S2),aTP_Lamp_sx(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_4899_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,S2: set(A),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S2),F3)
         => ( ( aa(A,B,F3,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S2),aTP_Lamp_sy(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_4900_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,X: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sz(fun(A,A),fun(A,fun(A,A)),F3),X),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_4901_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,X: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,X,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sz(fun(A,A),fun(A,fun(A,A)),F3),X),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_4902_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_ta(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_4903_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K2: real,F3: fun(A,B),K5: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K2))
         => ( ! [H3: A] :
                ( ( H3 != zero_zero(A) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K2))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,H3))),aa(real,real,aa(real,fun(real,real),times_times(real),K5),real_V7770717601297561774m_norm(A,H3)))) ) )
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_4904_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B2: real,F3: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
           => ? [M8: A] :
                ( ! [X5: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X5))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X5),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M8),aa(real,A,F3,X5))) )
                & ? [X3: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X3))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
                    & ( aa(real,A,F3,X3) = M8 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_4905_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B2: real,F3: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
           => ? [M8: A] :
                ( ! [X5: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X5))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X5),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F3,X5)),M8)) )
                & ? [X3: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X3))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
                    & ( aa(real,A,F3,X3) = M8 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_4906_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B2: real,F3: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
           => ? [M8: A] :
              ! [X5: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X5))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X5),B2)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F3,X5)),M8)) ) ) ) ) ).

% isCont_bounded
tff(fact_4907_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,X: A] :
          ( has_derivative(A,A,F3,aa(A,fun(A,A),times_times(A),D5),topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_sz(fun(A,A),fun(A,fun(A,A)),F3),X),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_4908_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G3)
           => ( ( aa(A,B,G3,A3) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_sv(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ) ).

% isCont_divide
tff(fact_4909_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F3)
         => ( ( aa(A,B,F3,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_sy(fun(A,B),fun(A,B),F3)) ) ) ) ).

% isCont_sgn
tff(fact_4910_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(A,B),F4: filter(B),A3: A] :
          ( filterlim(A,B,F3,F4,topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_tb(fun(A,B),fun(A,fun(A,B)),F3),A3),F4,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_4911_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S2: set(A),F3: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S2),F3)
         => ( ( cos(A,aa(A,A,F3,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S2),aTP_Lamp_rn(fun(A,A),fun(A,A),F3)) ) ) ) ).

% continuous_within_tan
tff(fact_4912_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S2: set(A),F3: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S2),F3)
         => ( ( sin(A,aa(A,A,F3,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S2),aTP_Lamp_sn(fun(A,A),fun(A,A),F3)) ) ) ) ).

% continuous_within_cot
tff(fact_4913_continuous__at__within__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: C,A6: set(C),F3: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,A6),F3)
         => ( ( cosh(A,aa(C,A,F3,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,A6),aTP_Lamp_tc(fun(C,A),fun(C,A),F3)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_4914_CARAT__DERIV,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),L: A,X: A] :
          ( has_field_derivative(A,F3,L,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ? [G6: fun(A,A)] :
              ( ! [Z3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,F3,Z3)),aa(A,A,F3,X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G6,Z3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z3),X))
              & topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),G6)
              & ( aa(A,A,G6,X) = L ) ) ) ) ).

% CARAT_DERIV
tff(fact_4915_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B2: real,F3: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
           => ? [M8: A] :
                ( ! [X5: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X5))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X5),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F3,X5)),M8)) )
                & ! [N8: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N8),M8))
                   => ? [X3: real] :
                        ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X3))
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N8),aa(real,A,F3,X3))) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_4916_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_4917_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_4918_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_4919_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S2: real,A3: fun(nat,A),F3: fun(A,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S2))
         => ( ! [X3: A] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X3)),S2))
               => sums(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),A3),X3),aa(A,A,F3,X3)) )
           => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,aa(nat,A,A3,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_4920_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S2: real,A3: fun(nat,A),F3: fun(A,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S2))
         => ( ! [X3: A] :
                ( ( X3 != zero_zero(A) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X3)),S2))
                 => sums(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),A3),X3),aa(A,A,F3,X3)) ) )
           => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,aa(nat,A,A3,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_4921_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K2: real,F3: fun(nat,real),G3: fun(A,fun(nat,B))] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K2))
         => ( summable(real,F3)
           => ( ! [H3: A,N3: nat] :
                  ( ( H3 != zero_zero(A) )
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K2))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G3,H3),N3))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F3,N3)),real_V7770717601297561774m_norm(A,H3)))) ) )
             => filterlim(A,B,aTP_Lamp_td(fun(A,fun(nat,B)),fun(A,B),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% lemma_termdiff5
tff(fact_4922_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,F3: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),F3)
         => ( ( cos(A,aa(A,A,F3,A3)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_rn(fun(A,A),fun(A,A),F3)) ) ) ) ).

% isCont_tan'
tff(fact_4923_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,F3: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),F3)
         => ( ( sin(A,aa(A,A,F3,A3)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_sn(fun(A,A),fun(A,A),F3)) ) ) ) ).

% isCont_cot'
tff(fact_4924_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: A,C3: fun(nat,A),N: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_te(fun(nat,A),fun(nat,fun(A,A)),C3),N)) ) ).

% isCont_polynom
tff(fact_4925_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),X: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),C3),Y3))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_ok(fun(nat,A),fun(A,A),C3)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_4926_GMVT_H,axiom,
    ! [A3: real,B2: real,F3: fun(real,real),G3: fun(real,real),G5: fun(real,real),F6: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),B2))
     => ( ! [Z: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),Z))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z),B2))
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z,top_top(set(real))),F3) ) )
       => ( ! [Z: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),Z))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z),B2))
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z,top_top(set(real))),G3) ) )
         => ( ! [Z: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),Z))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z),B2))
                 => has_field_derivative(real,G3,aa(real,real,G5,Z),topolo174197925503356063within(real,Z,top_top(set(real)))) ) )
           => ( ! [Z: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),Z))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z),B2))
                   => has_field_derivative(real,F3,aa(real,real,F6,Z),topolo174197925503356063within(real,Z,top_top(set(real)))) ) )
             => ? [C2: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),C2))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),B2))
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,A3))),aa(real,real,G5,C2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,G3,B2)),aa(real,real,G3,A3))),aa(real,real,F6,C2)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_4927_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K5: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K5)))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_ok(fun(nat,A),fun(A,A),C3)) ) ) ) ).

% isCont_powser
tff(fact_4928_isCont__powser_H,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa)
        & topological_t2_space(A) )
     => ! [A3: A,F3: fun(A,Aa),C3: fun(nat,Aa),K5: Aa] :
          ( topolo3448309680560233919inuous(A,Aa,topolo174197925503356063within(A,A3,top_top(set(A))),F3)
         => ( summable(Aa,aa(Aa,fun(nat,Aa),aTP_Lamp_tf(fun(nat,Aa),fun(Aa,fun(nat,Aa)),C3),K5))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(Aa,aa(A,Aa,F3,A3))),real_V7770717601297561774m_norm(Aa,K5)))
             => topolo3448309680560233919inuous(A,Aa,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(nat,Aa),fun(A,Aa),aTP_Lamp_th(fun(A,Aa),fun(fun(nat,Aa),fun(A,Aa)),F3),C3)) ) ) ) ) ).

% isCont_powser'
tff(fact_4929_summable__Leibniz_I3_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,A3,zero_zero(nat))),zero_zero(real)))
         => ! [N4: nat] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),suminf(real,aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3))),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)),one_one(nat)))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_4930_summable__Leibniz_I2_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(nat,real,A3,zero_zero(nat))))
         => ! [N4: nat] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),suminf(real,aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3))),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)),one_one(nat))))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_4931_summable__Leibniz_H_I4_J,axiom,
    ! [A3: fun(nat,real),N: nat] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N3)))
       => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N3))),aa(nat,real,A3,N3)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),suminf(real,aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_4932_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A3: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_tj(A,fun(fun(nat,A),fun(nat,A)),C3),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_4933_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A3: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_tk(A,fun(fun(nat,A),fun(nat,A)),C3),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_4934_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A3: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_tl(A,fun(fun(nat,A),fun(nat,A)),C3),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_4935_LIMSEQ__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),L: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aTP_Lamp_tm(fun(nat,A),fun(nat,A),F3),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_4936_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_tm(fun(nat,A),fun(nat,A),F3),topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_imp_Suc
tff(fact_4937_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),X: A,A3: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N8: nat] :
              ! [N3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N3)),A3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3)) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_4938_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),X: A,A3: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N8: nat] :
              ! [N3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(nat,A,X6,N3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X)) ) ) ) ).

% LIMSEQ_le_const
tff(fact_4939_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(nat,A),L: A,N6: nat,C5: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C5),aa(nat,A,F3,N3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C5),L)) ) ) ) ).

% Lim_bounded2
tff(fact_4940_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(nat,A),L: A,M5: nat,C5: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N3)),C5)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),C5)) ) ) ) ).

% Lim_bounded
tff(fact_4941_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),X: A,Y6: fun(nat,A),Y: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
           => ( ? [N8: nat] :
                ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N3)),aa(nat,A,Y6,N3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ).

% LIMSEQ_le
tff(fact_4942_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N6: nat,X6: fun(nat,A),Y6: fun(nat,A),X: A,Y: A] :
          ( ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N3)),aa(nat,A,Y6,N3))) )
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ).

% lim_mono
tff(fact_4943_summable__LIMSEQ__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% summable_LIMSEQ_zero
tff(fact_4944_incseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),L5: A,N: nat] :
          ( pp(aa(fun(nat,A),bool,order_mono(nat,A),X6))
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N)),L5)) ) ) ) ).

% incseq_le
tff(fact_4945_mult__nat__left__at__top,axiom,
    ! [C3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C3))
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C3),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_4946_mult__nat__right__at__top,axiom,
    ! [C3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C3))
     => filterlim(nat,nat,aTP_Lamp_tn(nat,fun(nat,nat),C3),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_4947_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: fun(nat,A),X: A] :
          ( topological_monoseq(A,A3)
         => ( filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A3,N4)),X))
                & ! [M3: nat,N4: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N4))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A3,M3)),aa(nat,A,A3,N4))) ) )
              | ( ! [N4: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(nat,A,A3,N4)))
                & ! [M3: nat,N4: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N4))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A3,N4)),aa(nat,A,A3,M3))) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_4948_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A] : filterlim(nat,A,aTP_Lamp_to(A,fun(nat,A),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_4949_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_tp(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_4950_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X6: fun(nat,A),X: A,L: nat] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),L))
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_tq(fun(nat,A),fun(nat,fun(nat,A)),X6),L),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_4951_telescope__summable_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => summable(A,aTP_Lamp_tr(fun(nat,A),fun(nat,A),F3)) ) ) ).

% telescope_summable'
tff(fact_4952_telescope__summable,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => summable(A,aTP_Lamp_ts(fun(nat,A),fun(nat,A),F3)) ) ) ).

% telescope_summable
tff(fact_4953_nested__sequence__unique,axiom,
    ! [F3: fun(nat,real),G3: fun(nat,real)] :
      ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,N3)),aa(nat,real,F3,aa(nat,nat,suc,N3))))
     => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,G3,aa(nat,nat,suc,N3))),aa(nat,real,G3,N3)))
       => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,N3)),aa(nat,real,G3,N3)))
         => ( filterlim(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_tt(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F3),G3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L3: real] :
                ( ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,N4)),L3))
                & filterlim(nat,real,F3,topolo7230453075368039082e_nhds(real,L3),at_top(nat))
                & ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L3),aa(nat,real,G3,N4)))
                & filterlim(nat,real,G3,topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_4954_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun(nat,real)] :
      ( ! [R3: real] :
        ? [N8: nat] :
        ! [N3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N3))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),R3),aa(nat,real,X6,N3))) )
     => filterlim(nat,real,aTP_Lamp_tu(fun(nat,real),fun(nat,real),X6),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_4955_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_tv(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_4956_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_tw(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_4957_increasing__LIMSEQ,axiom,
    ! [F3: fun(nat,real),L: real] :
      ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,N3)),aa(nat,real,F3,aa(nat,nat,suc,N3))))
     => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,N3)),L))
       => ( ! [E2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
             => ? [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F3,N4)),E2))) )
         => filterlim(nat,real,F3,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_4958_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_tx(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_4959_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_ty(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_4960_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_tz(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_4961_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => sums(A,aTP_Lamp_tr(fun(nat,A),fun(nat,A),F3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),C3)) ) ) ).

% telescope_sums'
tff(fact_4962_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => sums(A,aTP_Lamp_ts(fun(nat,A),fun(nat,A),F3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_4963_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A),S2: A] :
          ( sums(A,F3,S2)
        <=> filterlim(nat,A,aTP_Lamp_ua(fun(nat,A),fun(nat,A),F3),topolo7230453075368039082e_nhds(A,S2),at_top(nat)) ) ) ).

% sums_def'
tff(fact_4964_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_ub(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_4965_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A,R: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
           => ? [No: nat] :
              ! [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No),N4))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,N4)),L5))),R)) ) ) ) ) ).

% LIMSEQ_D
tff(fact_4966_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
             => ? [No2: nat] :
                ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No2),N3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,N3)),L5))),R3)) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_4967_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No3: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No3),N5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X6,N5)),L5))),R5)) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_4968_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_4969_tendsto__power__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F3: fun(B,nat),F4: filter(B),X: A] :
          ( filterlim(B,nat,F3,at_top(nat),F4)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
           => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_uc(fun(B,nat),fun(A,fun(B,A)),F3),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ) ).

% tendsto_power_zero
tff(fact_4970_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_ud(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_4971_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N3))),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N3)))))
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_4972_summable__Leibniz_I1_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => summable(real,aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3)) ) ) ).

% summable_Leibniz(1)
tff(fact_4973_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Df: A,Z2: A,S2: fun(nat,A),A3: A] :
          ( has_field_derivative(A,F3,Df,topolo174197925503356063within(A,Z2,top_top(set(A))))
         => ( filterlim(nat,A,S2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N3: nat] : aa(nat,A,S2,N3) != zero_zero(A)
             => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ue(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F3),Z2),S2),topolo7230453075368039082e_nhds(A,A3),at_top(nat))
               => ( Df = A3 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_4974_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => filterlim(nat,A,aTP_Lamp_uf(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_4975_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,X)))
         => filterlim(nat,A,aTP_Lamp_ug(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_4976_summable,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N3)))
       => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N3))),aa(nat,real,A3,N3)))
         => summable(real,aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3)) ) ) ) ).

% summable
tff(fact_4977_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_uh(fun(nat,real),fun(real,fun(nat,real)),Theta),Theta2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ~ ! [K: fun(nat,int)] : ~ filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ui(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K),topolo7230453075368039082e_nhds(real,Theta2),at_top(nat)) ) ).

% cos_diff_limit_1
tff(fact_4978_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_uj(fun(nat,real),fun(nat,real),Theta),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ? [K: fun(nat,int)] : filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ui(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% cos_limit_1
tff(fact_4979_summable__Leibniz_I4_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => filterlim(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_4980_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => filterlim(nat,real,aTP_Lamp_gg(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_4981_summable__Leibniz_H_I2_J,axiom,
    ! [A3: fun(nat,real),N: nat] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N3)))
       => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N3))),aa(nat,real,A3,N3)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),suminf(real,aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_4982_summable__Leibniz_H_I3_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N3)))
       => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N3))),aa(nat,real,A3,N3)))
         => filterlim(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_4983_sums__alternating__upper__lower,axiom,
    ! [A3: fun(nat,real)] :
      ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N3))),aa(nat,real,A3,N3)))
     => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N3)))
       => ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L3: real] :
              ( ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)))),L3))
              & filterlim(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,L3),at_top(nat))
              & ! [N4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L3),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)),one_one(nat))))))
              & filterlim(nat,real,aTP_Lamp_ul(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_4984_summable__Leibniz_I5_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => filterlim(nat,real,aTP_Lamp_ul(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_4985_summable__Leibniz_H_I5_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N3)))
       => ( ! [N3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N3))),aa(nat,real,A3,N3)))
         => filterlim(nat,real,aTP_Lamp_ul(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ti(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_4986_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_mh(fun(nat,A),fun(nat,A),F3),F4,at_top(nat))
    <=> filterlim(nat,A,F3,F4,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_4987_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),X: A] :
          ( has_derivative(A,B,F3,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_um(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F6),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_4988_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),D5: fun(A,B),X: A] :
          ( has_derivative(A,B,F3,D5,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D5)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_un(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F3),D5),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_4989_bounded__linear__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_uo(A,fun(A,A),Y)) ) ).

% bounded_linear_mult_left
tff(fact_4990_bounded__linear__const__mult,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & real_V822414075346904944vector(C) )
     => ! [G3: fun(C,A),X: A] :
          ( real_V3181309239436604168linear(C,A,G3)
         => real_V3181309239436604168linear(C,A,aa(A,fun(C,A),aTP_Lamp_pv(fun(C,A),fun(A,fun(C,A)),G3),X)) ) ) ).

% bounded_linear_const_mult
tff(fact_4991_bounded__linear__mult__const,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & real_V822414075346904944vector(C) )
     => ! [G3: fun(C,A),Y: A] :
          ( real_V3181309239436604168linear(C,A,G3)
         => real_V3181309239436604168linear(C,A,aa(A,fun(C,A),aTP_Lamp_pw(fun(C,A),fun(A,fun(C,A)),G3),Y)) ) ) ).

% bounded_linear_mult_const
tff(fact_4992_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_4993_real__bounded__linear,axiom,
    ! [F3: fun(real,real)] :
      ( real_V3181309239436604168linear(real,real,F3)
    <=> ? [C4: real] :
        ! [X4: real] : aa(real,real,F3,X4) = aa(real,real,aa(real,fun(real,real),times_times(real),X4),C4) ) ).

% real_bounded_linear
tff(fact_4994_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_py(A,B)) ) ).

% bounded_linear_zero
tff(fact_4995_bounded__linear_Obounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K9: real] :
            ! [X5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X5))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X5)),K9))) ) ) ).

% bounded_linear.bounded
tff(fact_4996_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),G3: fun(C,A),F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ( filterlim(C,A,G3,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_up(fun(A,B),fun(fun(C,A),fun(C,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_4997_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K9: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K9))
              & ! [X5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X5))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X5)),K9))) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_4998_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K9: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K9))
              & ! [X5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X5))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X5)),K9))) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_4999_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),K5: real] :
          ( ! [X3: A,Y3: A] : aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))
         => ( ! [R3: real,X3: A] : aa(A,B,F3,aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),R3),X3)) = aa(B,B,aa(real,fun(B,B),real_V8093663219630862766scaleR(B),R3),aa(A,B,F3,X3))
           => ( ! [X3: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K5)))
             => real_V3181309239436604168linear(A,B,F3) ) ) ) ) ).

% bounded_linear_intro
tff(fact_5000_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_5001_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F6: fun(A,B),X: A,F3: fun(A,B),S2: 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_uq(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F6),X),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S2))
           => has_derivative(A,B,F3,F6,topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivativeI
tff(fact_5002_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),X: A,S2: set(A)] :
          ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,X,S2))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ur(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F6),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivative_at_within
tff(fact_5003_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),X: A] :
          ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & ? [E4: fun(A,B)] :
                ( ! [H4: A] : aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,X)),aa(A,B,F6,H4))),aa(A,B,E4,H4))
                & filterlim(A,real,aTP_Lamp_us(fun(A,B),fun(A,real),E4),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% has_derivative_iff_Ex
tff(fact_5004_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),X: A,S2: set(A)] :
          ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,X,S2))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_um(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F6),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% has_derivative_within
tff(fact_5005_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F3,F6,F4)
        <=> ( 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_uu(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F3),F6),F4),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% has_derivative_def
tff(fact_5006_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [X: A,S3: set(A),F3: fun(A,B),F6: fun(A,B)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S3))
         => ( topolo1002775350975398744n_open(A,S3)
           => ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,X,S3))
            <=> ( real_V3181309239436604168linear(A,B,F6)
                & ? [E4: fun(A,B)] :
                    ( ! [H4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H4)),S3))
                       => ( aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,X)),aa(A,B,F6,H4))),aa(A,B,E4,H4)) ) )
                    & filterlim(A,real,aTP_Lamp_us(fun(A,B),fun(A,real),E4),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ) ).

% has_derivative_at_within_iff_Ex
tff(fact_5007_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_uv(fun(A,A),fun(A,A),F3),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,L),at_infinity(A)) ) ) ).

% lim_zero_infinity
tff(fact_5008_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S6: set(A)] :
              ( topolo1002775350975398744n_open(A,S6)
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),F0),S6))
               => ? [N7: nat] :
                  ! [N5: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N5))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,F3,N5)),S6)) ) ) ) ) ) ).

% lim_explicit
tff(fact_5009_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,B,F4,G3)
           => ( ( aa(A,B,G3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_uw(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_sv(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ) ).

% continuous_divide
tff(fact_5010_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F4: filter(A),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( ( aa(A,B,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_uw(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_sx(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_inverse
tff(fact_5011_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( ( aa(A,B,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_uw(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_sy(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_sgn
tff(fact_5012_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_5013_tendsto__mult__filterlim__at__infinity,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(B,A),C3: A,F4: filter(B),G3: fun(B,A)] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,C3),F4)
         => ( ( C3 != zero_zero(A) )
           => ( filterlim(B,A,G3,at_infinity(A),F4)
             => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ux(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3),at_infinity(A),F4) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_5014_tendsto__divide__0,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(C,A),C3: A,F4: filter(C),G3: fun(C,A)] :
          ( filterlim(C,A,F3,topolo7230453075368039082e_nhds(A,C3),F4)
         => ( filterlim(C,A,G3,at_infinity(A),F4)
           => filterlim(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_uy(fun(C,A),fun(fun(C,A),fun(C,A)),F3),G3),topolo7230453075368039082e_nhds(A,zero_zero(A)),F4) ) ) ) ).

% tendsto_divide_0
tff(fact_5015_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),N: nat] :
          ( filterlim(A,B,F3,at_infinity(B),F4)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_uz(fun(A,B),fun(nat,fun(A,B)),F3),N),at_infinity(B),F4) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_5016_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F3: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F3)
         => ( ( cos(A,aa(A,A,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_va(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_rn(fun(A,A),fun(A,A),F3)) ) ) ) ).

% continuous_tan
tff(fact_5017_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F3: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F3)
         => ( ( sin(A,aa(A,A,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_va(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_sn(fun(A,A),fun(A,A),F3)) ) ) ) ).

% continuous_cot
tff(fact_5018_continuous__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(C),F3: fun(C,A)] :
          ( topolo3448309680560233919inuous(C,A,F4,F3)
         => ( ( cosh(A,aa(C,A,F3,topolo3827282254853284352ce_Lim(C,C,F4,aTP_Lamp_vb(C,C)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(C,A,F4,aTP_Lamp_tc(fun(C,A),fun(C,A),F3)) ) ) ) ).

% continuous_tanh
tff(fact_5019_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_5020_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_vc(fun(A,B),fun(A,B),G3),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F4)
        <=> filterlim(A,B,G3,at_infinity(B),F4) ) ) ).

% filterlim_inverse_at_iff
tff(fact_5021_tendsto__offset__zero__iff,axiom,
    ! [C: $tType,D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D)
        & zero(C) )
     => ! [A3: A,S3: set(A),F3: fun(A,D),L5: D] :
          ( nO_MATCH(C,A,zero_zero(C),A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),S3))
           => ( topolo1002775350975398744n_open(A,S3)
             => ( filterlim(A,D,F3,topolo7230453075368039082e_nhds(D,L5),topolo174197925503356063within(A,A3,S3))
              <=> filterlim(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_ss(A,fun(fun(A,D),fun(A,D)),A3),F3),topolo7230453075368039082e_nhds(D,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_5022_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),C3: A,F4: filter(A),G3: fun(A,A)] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,C3),F4)
         => ( filterlim(A,A,G3,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F4)
           => ( ( C3 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_pe(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),at_infinity(A),F4) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_5023_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C3: fun(nat,A),K2: nat,N: nat,B6: real] :
          ( ( aa(nat,A,C3,K2) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
             => eventually(A,aa(real,fun(A,bool),aa(nat,fun(real,fun(A,bool)),aTP_Lamp_vd(fun(nat,A),fun(nat,fun(real,fun(A,bool))),C3),N),B6),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_5024_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_ve(real,fun(real,real),X),topolo7230453075368039082e_nhds(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_5025_GMVT,axiom,
    ! [A3: real,B2: real,F3: fun(real,real),G3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),B2))
     => ( ! [X3: real] :
            ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X3))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
       => ( ! [X3: real] :
              ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),X3))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2)) )
             => differentiable(real,real,F3,topolo174197925503356063within(real,X3,top_top(set(real)))) )
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A3),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),G3) )
           => ( ! [X3: real] :
                  ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),X3))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2)) )
                 => differentiable(real,real,G3,topolo174197925503356063within(real,X3,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( has_field_derivative(real,G3,G_c,topolo174197925503356063within(real,C2,top_top(set(real))))
                  & has_field_derivative(real,F3,F_c,topolo174197925503356063within(real,C2,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),C2))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),B2))
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,A3))),G_c) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,G3,B2)),aa(real,real,G3,A3))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_5026_greaterThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,set(A),set_ord_greaterThan(A),X) = aa(A,set(A),set_ord_greaterThan(A),Y) )
        <=> ( X = Y ) ) ) ).

% greaterThan_eq_iff
tff(fact_5027_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),aa(A,set(A),set_ord_greaterThan(A),K2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K2),I2)) ) ) ).

% greaterThan_iff
tff(fact_5028_eventually__sequentially__Suc,axiom,
    ! [P2: fun(nat,bool)] :
      ( eventually(nat,aTP_Lamp_iy(fun(nat,bool),fun(nat,bool),P2),at_top(nat))
    <=> eventually(nat,P2,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_5029_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),X)),aa(A,set(A),set_ord_greaterThan(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% greaterThan_subset_iff
tff(fact_5030_Compl__greaterThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K2: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_greaterThan(A),K2)) = aa(A,set(A),set_ord_atMost(A),K2) ) ).

% Compl_greaterThan
tff(fact_5031_Compl__atMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K2: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_atMost(A),K2)) = aa(A,set(A),set_ord_greaterThan(A),K2) ) ).

% Compl_atMost
tff(fact_5032_image__uminus__lessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : image2(A,A,uminus_uminus(A),aa(A,set(A),set_ord_lessThan(A),X)) = aa(A,set(A),set_ord_greaterThan(A),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_lessThan
tff(fact_5033_image__uminus__greaterThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : image2(A,A,uminus_uminus(A),aa(A,set(A),set_ord_greaterThan(A),X)) = aa(A,set(A),set_ord_lessThan(A),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_greaterThan
tff(fact_5034_differentiable__cmult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [C3: A,Q2: fun(B,A),T2: B] :
          ( differentiable(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vf(A,fun(fun(B,A),fun(B,A)),C3),Q2),topolo174197925503356063within(B,T2,top_top(set(B))))
        <=> ( ( C3 = zero_zero(A) )
            | differentiable(B,A,Q2,topolo174197925503356063within(B,T2,top_top(set(B)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_5035_differentiable__cmult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [Q2: fun(B,A),C3: A,T2: B] :
          ( differentiable(B,A,aa(A,fun(B,A),aTP_Lamp_vg(fun(B,A),fun(A,fun(B,A)),Q2),C3),topolo174197925503356063within(B,T2,top_top(set(B))))
        <=> ( ( C3 = zero_zero(A) )
            | differentiable(B,A,Q2,topolo174197925503356063within(B,T2,top_top(set(B)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_5036_greaterThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] : aa(A,set(A),set_ord_greaterThan(A),X) != bot_bot(set(A)) ) ).

% greaterThan_non_empty
tff(fact_5037_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_greaterThan(A),L) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_5038_infinite__Ioi,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [A3: A] : ~ pp(aa(set(A),bool,finite_finite2(A),aa(A,set(A),set_ord_greaterThan(A),A3))) ) ).

% infinite_Ioi
tff(fact_5039_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,bool)] :
          ( eventually(A,P2,at_top(A))
        <=> ? [N7: A] :
            ! [N5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N7),N5))
             => pp(aa(A,bool,P2,N5)) ) ) ) ).

% eventually_at_top_linorder
tff(fact_5040_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,P2: fun(A,bool)] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),X3))
             => pp(aa(A,bool,P2,X3)) )
         => eventually(A,P2,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_5041_eventually__sequentially,axiom,
    ! [P2: fun(nat,bool)] :
      ( eventually(nat,P2,at_top(nat))
    <=> ? [N7: nat] :
        ! [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N5))
         => pp(aa(nat,bool,P2,N5)) ) ) ).

% eventually_sequentially
tff(fact_5042_eventually__sequentiallyI,axiom,
    ! [C3: nat,P2: fun(nat,bool)] :
      ( ! [X3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C3),X3))
         => pp(aa(nat,bool,P2,X3)) )
     => eventually(nat,P2,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_5043_lessThan__Int__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),A3)),aa(A,set(A),set_ord_greaterThan(A),B2)) = aa(A,set(A),set_ord_greaterThan(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) ) ).

% lessThan_Int_lessThan
tff(fact_5044_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : eventually(A,aa(A,fun(A,bool),ord_less_eq(A),C3),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_5045_le__sequentially,axiom,
    ! [F4: filter(nat)] :
      ( pp(aa(filter(nat),bool,aa(filter(nat),fun(filter(nat),bool),ord_less_eq(filter(nat)),F4),at_top(nat)))
    <=> ! [N7: nat] : eventually(nat,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),F4) ) ).

% le_sequentially
tff(fact_5046_differentiable__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F3: fun(A,B),X: A,S2: set(A),G3: fun(A,B)] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,X,S2))
         => ( differentiable(A,B,G3,topolo174197925503356063within(A,X,S2))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vh(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% differentiable_mult
tff(fact_5047_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F3: fun(A,B),P2: fun(B,bool),G3: fun(B,A)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P2,X3))
               => ( aa(A,B,F3,aa(B,A,G3,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P2,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G3,X3))) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P2,at_top(B))
                 => filterlim(A,B,F3,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_5048_ivl__disj__int__one_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(7)
tff(fact_5049_tendsto__le,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F4: filter(B),F3: fun(B,A),X: A,G3: fun(B,A),Y: A] :
          ( ( F4 != bot_bot(filter(B)) )
         => ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,X),F4)
           => ( filterlim(B,A,G3,topolo7230453075368039082e_nhds(A,Y),F4)
             => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_vi(fun(B,A),fun(fun(B,A),fun(B,bool)),F3),G3),F4)
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ) ) ).

% tendsto_le
tff(fact_5050_tendsto__lowerbound,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(B,A),X: A,F4: filter(B),A3: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,X),F4)
         => ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_vj(fun(B,A),fun(A,fun(B,bool)),F3),A3),F4)
           => ( ( F4 != bot_bot(filter(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X)) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_5051_tendsto__upperbound,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(B,A),X: A,F4: filter(B),A3: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,X),F4)
         => ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_vk(fun(B,A),fun(A,fun(B,bool)),F3),A3),F4)
           => ( ( F4 != bot_bot(filter(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3)) ) ) ) ) ).

% tendsto_upperbound
tff(fact_5052_tendsto__sandwich,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F3: fun(B,A),G3: fun(B,A),Net: filter(B),H: fun(B,A),C3: A] :
          ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_vl(fun(B,A),fun(fun(B,A),fun(B,bool)),F3),G3),Net)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_vl(fun(B,A),fun(fun(B,A),fun(B,bool)),G3),H),Net)
           => ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,C3),Net)
             => ( filterlim(B,A,H,topolo7230453075368039082e_nhds(A,C3),Net)
               => filterlim(B,A,G3,topolo7230453075368039082e_nhds(A,C3),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_5053_ivl__disj__un__one_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_5054_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_vm(fun(A,B),fun(B,fun(A,bool)),F3),Z6),F4) ) ) ).

% filterlim_at_top
tff(fact_5055_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z6: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C3),Z6))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_vm(fun(A,B),fun(B,fun(A,bool)),F3),Z6),F4) ) ) ) ).

% filterlim_at_top_ge
tff(fact_5056_filterlim__at__top__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),F4: filter(B),G3: fun(B,A)] :
          ( filterlim(B,A,F3,at_top(A),F4)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_vn(fun(B,A),fun(fun(B,A),fun(B,bool)),F3),G3),F4)
           => filterlim(B,A,G3,at_top(A),F4) ) ) ) ).

% filterlim_at_top_mono
tff(fact_5057_ivl__disj__int__one_I5_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(5)
tff(fact_5058_greaterThanLessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or5935395276787703475ssThan(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),L)),aa(A,set(A),set_ord_lessThan(A),U)) ) ).

% greaterThanLessThan_def
tff(fact_5059_greaterThanLessThan__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A] : set_or5935395276787703475ssThan(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),A3)),aa(A,set(A),set_ord_lessThan(A),B2)) ) ).

% greaterThanLessThan_eq
tff(fact_5060_greaterThanAtMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or3652927894154168847AtMost(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),L)),aa(A,set(A),set_ord_atMost(A),U)) ) ).

% greaterThanAtMost_def
tff(fact_5061_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),X: A,S2: set(A),G3: fun(A,B)] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,X,S2))
         => ( differentiable(A,B,G3,topolo174197925503356063within(A,X,S2))
           => ( ( aa(A,B,G3,X) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vo(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% differentiable_divide
tff(fact_5062_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),X: A,S2: set(A)] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,X,S2))
         => ( ( aa(A,B,F3,X) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_vp(fun(A,B),fun(A,B),F3),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% differentiable_inverse
tff(fact_5063_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P2: fun(A,bool),A3: A] :
          ( eventually(A,P2,topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,bool),aTP_Lamp_vq(fun(A,bool),fun(A,fun(A,bool)),P2),A3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_5064_increasing__tendsto,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F3: fun(B,A),L: A,F4: filter(B)] :
          ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_vr(fun(B,A),fun(A,fun(B,bool)),F3),L),F4)
         => ( ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),L))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_vs(fun(B,A),fun(A,fun(B,bool)),F3),X3),F4) )
           => filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% increasing_tendsto
tff(fact_5065_decreasing__tendsto,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [L: A,F3: fun(B,A),F4: filter(B)] :
          ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_vt(A,fun(fun(B,A),fun(B,bool)),L),F3),F4)
         => ( ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),X3))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_vu(fun(B,A),fun(A,fun(B,bool)),F3),X3),F4) )
           => filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% decreasing_tendsto
tff(fact_5066_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z6: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),C3),Z6))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_vv(fun(A,B),fun(B,fun(A,bool)),F3),Z6),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_5067_filterlim__times__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(B,A),P: A,F12: filter(B),C3: A,L: A] :
          ( filterlim(B,A,F3,topolo174197925503356063within(A,P,aa(A,set(A),set_ord_greaterThan(A),P)),F12)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
           => ( ( L = aa(A,A,aa(A,fun(A,A),times_times(A),C3),P) )
             => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_vw(fun(B,A),fun(A,fun(B,A)),F3),C3),topolo174197925503356063within(A,L,aa(A,set(A),set_ord_greaterThan(A),L)),F12) ) ) ) ) ).

% filterlim_times_pos
tff(fact_5068_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F3: fun(A,B),P2: fun(B,bool),G3: fun(B,A),A3: A] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P2,X3))
               => ( aa(A,B,F3,aa(B,A,G3,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P2,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G3,X3))) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_lessThan(A),A3)))
               => ( ! [B4: A] :
                      ( pp(aa(A,bool,Q,B4))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),A3)) )
                 => ( eventually(B,P2,at_top(B))
                   => filterlim(A,B,F3,at_top(B),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_lessThan(A),A3))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_5069_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,C),K5: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( eventually(A,aa(real,fun(A,bool),aa(fun(A,C),fun(real,fun(A,bool)),aTP_Lamp_vx(fun(A,B),fun(fun(A,C),fun(real,fun(A,bool))),F3),G3),K5),F4)
           => filterlim(A,C,G3,topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% tendsto_0_le
tff(fact_5070_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,G3: fun(A,B),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_lessThan(A),A3)),G3)
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,G3,A3)),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_greaterThan(A),A3)))
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_vy(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),G3),F3)) ) ) ) ).

% isCont_If_ge
tff(fact_5071_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_vz(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F3),G3),at_top(nat))
         => ( filterlim(nat,real,G3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F3) ) ) ) ).

% summable_bounded_partials
tff(fact_5072_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S3: set(A)] :
          ( ! [A5: A,B4: A,X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),S3))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),S3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B4))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3)) ) ) ) )
         => ? [A5: A,B4: A] :
              ( ( S3 = bot_bot(set(A)) )
              | ( S3 = top_top(set(A)) )
              | ( S3 = aa(A,set(A),set_ord_lessThan(A),B4) )
              | ( S3 = aa(A,set(A),set_ord_atMost(A),B4) )
              | ( S3 = aa(A,set(A),set_ord_greaterThan(A),A5) )
              | ( S3 = aa(A,set(A),set_ord_atLeast(A),A5) )
              | ( S3 = set_or5935395276787703475ssThan(A,A5,B4) )
              | ( S3 = set_or3652927894154168847AtMost(A,A5,B4) )
              | ( S3 = set_or7035219750837199246ssThan(A,A5,B4) )
              | ( S3 = set_or1337092689740270186AtMost(A,A5,B4) ) ) ) ) ).

% interval_cases
tff(fact_5073_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_wa(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F3),G3),at_top(nat))
         => ( filterlim(nat,real,G3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F3) ) ) ) ).

% summable_Cauchy'
tff(fact_5074_atLeast__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,set(A),set_ord_atLeast(A),X) = aa(A,set(A),set_ord_atLeast(A),Y) )
        <=> ( X = Y ) ) ) ).

% atLeast_eq_iff
tff(fact_5075_atLeast__0,axiom,
    aa(nat,set(nat),set_ord_atLeast(nat),zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_5076_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),aa(A,set(A),set_ord_atLeast(A),K2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K2),I2)) ) ) ).

% atLeast_iff
tff(fact_5077_atLeast__empty__triv,axiom,
    ! [A: $tType] : aa(set(A),set(set(A)),set_ord_atLeast(set(A)),bot_bot(set(A))) = top_top(set(set(A))) ).

% atLeast_empty_triv
tff(fact_5078_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),X)),aa(A,set(A),set_ord_atLeast(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% atLeast_subset_iff
tff(fact_5079_image__add__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A] : image2(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,set(A),set_ord_atLeast(A),I2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),I2)) ) ).

% image_add_atLeast
tff(fact_5080_Compl__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K2: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_lessThan(A),K2)) = aa(A,set(A),set_ord_atLeast(A),K2) ) ).

% Compl_lessThan
tff(fact_5081_Compl__atLeast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K2: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_atLeast(A),K2)) = aa(A,set(A),set_ord_lessThan(A),K2) ) ).

% Compl_atLeast
tff(fact_5082_Icc__subset__Ici__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,L2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atLeast(A),L2)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L2),L)) ) ) ) ).

% Icc_subset_Ici_iff
tff(fact_5083_image__minus__const__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A3: A] : image2(A,A,aa(A,fun(A,A),minus_minus(A),C3),aa(A,set(A),set_ord_atLeast(A),A3)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A3)) ) ).

% image_minus_const_atLeast
tff(fact_5084_image__minus__const__AtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] : image2(A,A,aa(A,fun(A,A),minus_minus(A),C3),aa(A,set(A),set_ord_atMost(A),B2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)) ) ).

% image_minus_const_AtMost
tff(fact_5085_image__uminus__atLeast,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : image2(A,A,uminus_uminus(A),aa(A,set(A),set_ord_atLeast(A),X)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_atLeast
tff(fact_5086_image__uminus__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : image2(A,A,uminus_uminus(A),aa(A,set(A),set_ord_atMost(A),X)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_atMost
tff(fact_5087_Int__atLeastAtMostL2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(A,set(A),set_ord_atLeast(A),C3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3),B2) ) ).

% Int_atLeastAtMostL2
tff(fact_5088_Int__atLeastAtMostR2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atLeast(A),A3)),set_or1337092689740270186AtMost(A,C3,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3),D3) ) ).

% Int_atLeastAtMostR2
tff(fact_5089_atLeast__Suc__greaterThan,axiom,
    ! [K2: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K2)) = aa(nat,set(nat),set_ord_greaterThan(nat),K2) ).

% atLeast_Suc_greaterThan
tff(fact_5090_infinite__Ici,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [A3: A] : ~ pp(aa(set(A),bool,finite_finite2(A),aa(A,set(A),set_ord_atLeast(A),A3))) ) ).

% infinite_Ici
tff(fact_5091_not__Iic__eq__Ici,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A,L2: A] : aa(A,set(A),set_ord_atMost(A),H) != aa(A,set(A),set_ord_atLeast(A),L2) ) ).

% not_Iic_eq_Ici
tff(fact_5092_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_atLeast(A),L) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_5093_not__empty__eq__Ici__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atLeast(A),L) ) ).

% not_empty_eq_Ici_eq_empty
tff(fact_5094_not__UNIV__eq__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L2: A] : top_top(set(A)) != aa(A,set(A),set_ord_atLeast(A),L2) ) ).

% not_UNIV_eq_Ici
tff(fact_5095_not__Ici__eq__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L2: A,L: A,H: A] : aa(A,set(A),set_ord_atLeast(A),L2) != set_or1337092689740270186AtMost(A,L,H) ) ).

% not_Ici_eq_Icc
tff(fact_5096_atLeast__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] :
          ( ( aa(A,set(A),set_ord_atLeast(A),X) = top_top(set(A)) )
        <=> ( X = bot_bot(A) ) ) ) ).

% atLeast_eq_UNIV_iff
tff(fact_5097_not__UNIV__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atLeast(A),L))) ) ).

% not_UNIV_le_Ici
tff(fact_5098_not__Ici__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,L2: A,H2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),set_or1337092689740270186AtMost(A,L2,H2))) ) ).

% not_Ici_le_Icc
tff(fact_5099_not__Ici__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_atMost(A),H2))) ) ).

% not_Ici_le_Iic
tff(fact_5100_not__Iic__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),aa(A,set(A),set_ord_atLeast(A),L2))) ) ).

% not_Iic_le_Ici
tff(fact_5101_Ioi__le__Ico,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),A3)),aa(A,set(A),set_ord_atLeast(A),A3))) ) ).

% Ioi_le_Ico
tff(fact_5102_filterlim__at__top__mult__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,at_top(real),F4)
     => ( filterlim(A,real,G3,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wb(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_top(real),F4) ) ) ).

% filterlim_at_top_mult_at_top
tff(fact_5103_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,bool)] :
          ( eventually(A,P2,at_top(A))
         => eventually(A,aTP_Lamp_wc(fun(A,bool),fun(A,bool),P2),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_5104_ivl__disj__un__one_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_5105_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),A3)),aa(A,set(A),set_ord_greaterThan(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_5106_ivl__disj__int__one_I8_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(8)
tff(fact_5107_atLeastLessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or7035219750837199246ssThan(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_lessThan(A),U)) ) ).

% atLeastLessThan_def
tff(fact_5108_atLeastAtMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or1337092689740270186AtMost(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_atMost(A),U)) ) ).

% atLeastAtMost_def
tff(fact_5109_ivl__disj__int__one_I6_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(6)
tff(fact_5110_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P2: fun(A,bool)] : ord_Least(A,P2) = the(A,aTP_Lamp_wd(fun(A,bool),fun(A,bool),P2)) ) ).

% Least_def
tff(fact_5111_greaterThan__0,axiom,
    aa(nat,set(nat),set_ord_greaterThan(nat),zero_zero(nat)) = image2(nat,nat,suc,top_top(set(nat))) ).

% greaterThan_0
tff(fact_5112_filterlim__pow__at__top,axiom,
    ! [A: $tType,N: nat,F3: fun(A,real),F4: filter(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(A,real,F3,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rm(nat,fun(fun(A,real),fun(A,real)),N),F3),at_top(real),F4) ) ) ).

% filterlim_pow_at_top
tff(fact_5113_atMost__Int__atLeast,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [N: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),N)),aa(A,set(A),set_ord_atLeast(A),N)) = aa(set(A),set(A),insert(A,N),bot_bot(set(A))) ) ).

% atMost_Int_atLeast
tff(fact_5114_ivl__disj__un__one_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_5115_ivl__disj__un__singleton_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,L),bot_bot(set(A)))),aa(A,set(A),set_ord_greaterThan(A),L)) = aa(A,set(A),set_ord_atLeast(A),L) ) ).

% ivl_disj_un_singleton(1)
tff(fact_5116_ivl__disj__un__one_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_5117_greaterThan__Suc,axiom,
    ! [K2: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K2)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_greaterThan(nat),K2)),aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K2)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_5118_atLeast__Suc,axiom,
    ! [K2: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K2)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atLeast(nat),K2)),aa(set(nat),set(nat),insert(nat,K2),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_5119_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C3))
       => ( filterlim(A,real,G3,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wb(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_top(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_5120_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C3))
       => ( filterlim(A,real,G3,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_we(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_top(real),F4) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_5121_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool)] : order_Greatest(A,P2) = the(A,aTP_Lamp_wf(fun(A,bool),fun(A,bool),P2)) ) ).

% Greatest_def
tff(fact_5122_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F3: fun(A,B),P2: fun(B,bool),G3: fun(B,A),A3: A] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P2,X3))
               => ( aa(A,B,F3,aa(B,A,G3,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P2,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G3,X3))) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_greaterThan(A),A3)))
               => ( ! [B4: A] :
                      ( pp(aa(A,bool,Q,B4))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B4)) )
                 => ( eventually(B,P2,at_bot(B))
                   => filterlim(A,B,F3,at_bot(B),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_greaterThan(A),A3))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_5123_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F3: fun(real,real),F4: filter(real)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(real,real,F3,at_bot(real),F4)
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wg(nat,fun(fun(real,real),fun(real,real)),N),F3),at_top(real),F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_5124_GreatestI__ex__nat,axiom,
    ! [P2: fun(nat,bool),B2: nat] :
      ( ? [X_1: nat] : pp(aa(nat,bool,P2,X_1))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => pp(aa(nat,bool,P2,order_Greatest(nat,P2))) ) ) ).

% GreatestI_ex_nat
tff(fact_5125_Greatest__le__nat,axiom,
    ! [P2: fun(nat,bool),K2: nat,B2: nat] :
      ( pp(aa(nat,bool,P2,K2))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),order_Greatest(nat,P2))) ) ) ).

% Greatest_le_nat
tff(fact_5126_GreatestI__nat,axiom,
    ! [P2: fun(nat,bool),K2: nat,B2: nat] :
      ( pp(aa(nat,bool,P2,K2))
     => ( ! [Y3: nat] :
            ( pp(aa(nat,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),B2)) )
       => pp(aa(nat,bool,P2,order_Greatest(nat,P2))) ) ) ).

% GreatestI_nat
tff(fact_5127_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,bool)] :
          ( eventually(A,P2,at_bot(A))
        <=> ? [N7: A] :
            ! [N5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N5),N7))
             => pp(aa(A,bool,P2,N5)) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_5128_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P2,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P2,Y3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( order_Greatest(A,P2) = X ) ) ) ) ).

% Greatest_equality
tff(fact_5129_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,bool),X: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P2,X))
         => ( ! [Y3: A] :
                ( pp(aa(A,bool,P2,Y3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P2,X3))
                 => ( ! [Y4: A] :
                        ( pp(aa(A,bool,P2,Y4))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X3)) )
                   => pp(aa(A,bool,Q,X3)) ) )
             => pp(aa(A,bool,Q,order_Greatest(A,P2))) ) ) ) ) ).

% GreatestI2_order
tff(fact_5130_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : eventually(A,aa(A,fun(A,bool),aTP_Lamp_lw(A,fun(A,bool)),C3),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_5131_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_wh(fun(A,B),fun(B,fun(A,bool)),F3),Z6),F4) ) ) ).

% filterlim_at_bot
tff(fact_5132_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Z6),C3))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_wh(fun(A,B),fun(B,fun(A,bool)),F3),Z6),F4) ) ) ) ).

% filterlim_at_bot_le
tff(fact_5133_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C3))
       => ( filterlim(A,real,G3,at_bot(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wb(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_5134_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C3),zero_zero(real)))
       => ( filterlim(A,real,G3,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wb(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_5135_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z6),C3))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_wi(fun(A,B),fun(B,fun(A,bool)),F3),Z6),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_5136_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F3: fun(real,real),F4: filter(real)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(real,real,F3,at_bot(real),F4)
       => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wg(nat,fun(fun(real,real),fun(real,real)),N),F3),at_bot(real),F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_5137_Bfun__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(B,A),A3: A,F4: filter(B)] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( ( A3 != zero_zero(A) )
           => bfun(B,A,aTP_Lamp_sk(fun(B,A),fun(B,A),F3),F4) ) ) ) ).

% Bfun_inverse
tff(fact_5138_MVT,axiom,
    ! [A3: real,B2: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B2),F3)
       => ( ! [X3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),X3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
               => differentiable(real,real,F3,topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ? [L3: real,Z: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A3),Z))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z),B2))
              & has_field_derivative(real,F3,L3,topolo174197925503356063within(real,Z,top_top(set(real))))
              & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A3)),L3) ) ) ) ) ) ).

% MVT
tff(fact_5139_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),P2: fun(A,bool)] : aa(fun(A,bool),A,least(A,Less_eq),P2) = the(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_wj(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Less_eq),P2)) ).

% ord.Least_def
tff(fact_5140_continuous__on__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [S2: set(A),F3: fun(A,B),G3: fun(A,C)] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => ( topolo81223032696312382ous_on(A,C,S2,G3)
           => topolo81223032696312382ous_on(A,product_prod(B,C),S2,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_wk(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3)) ) ) ) ).

% continuous_on_Pair
tff(fact_5141_ord_OLeast_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : least(A,Less_eq) = least(A,Less_eq) ).

% ord.Least.cong
tff(fact_5142_continuous__on__mult__const,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [S2: set(A),C3: A] : topolo81223032696312382ous_on(A,A,S2,aa(A,fun(A,A),times_times(A),C3)) ) ).

% continuous_on_mult_const
tff(fact_5143_continuous__on__mult,axiom,
    ! [A: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [S2: set(D),F3: fun(D,A),G3: fun(D,A)] :
          ( topolo81223032696312382ous_on(D,A,S2,F3)
         => ( topolo81223032696312382ous_on(D,A,S2,G3)
           => topolo81223032696312382ous_on(D,A,S2,aa(fun(D,A),fun(D,A),aTP_Lamp_wl(fun(D,A),fun(fun(D,A),fun(D,A)),F3),G3)) ) ) ) ).

% continuous_on_mult
tff(fact_5144_continuous__on__mult_H,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & topolo4211221413907600880p_mult(B) )
     => ! [A6: set(D),F3: fun(D,B),G3: fun(D,B)] :
          ( topolo81223032696312382ous_on(D,B,A6,F3)
         => ( topolo81223032696312382ous_on(D,B,A6,G3)
           => topolo81223032696312382ous_on(D,B,A6,aa(fun(D,B),fun(D,B),aTP_Lamp_wm(fun(D,B),fun(fun(D,B),fun(D,B)),F3),G3)) ) ) ) ).

% continuous_on_mult'
tff(fact_5145_continuous__on__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topolo4958980785337419405_space(B) )
     => ! [S2: set(B),F3: fun(B,A),C3: A] :
          ( topolo81223032696312382ous_on(B,A,S2,F3)
         => topolo81223032696312382ous_on(B,A,S2,aa(A,fun(B,A),aTP_Lamp_wn(fun(B,A),fun(A,fun(B,A)),F3),C3)) ) ) ).

% continuous_on_mult_left
tff(fact_5146_continuous__on__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topolo4958980785337419405_space(B) )
     => ! [S2: set(B),F3: fun(B,A),C3: A] :
          ( topolo81223032696312382ous_on(B,A,S2,F3)
         => topolo81223032696312382ous_on(B,A,S2,aa(A,fun(B,A),aTP_Lamp_wo(fun(B,A),fun(A,fun(B,A)),F3),C3)) ) ) ).

% continuous_on_mult_right
tff(fact_5147_IVT_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F3: fun(A,B),A3: A,Y: B,B2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,A3)),Y))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),aa(A,B,F3,B2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
             => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B2),F3)
               => ? [X3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
                    & ( aa(A,B,F3,X3) = Y ) ) ) ) ) ) ) ).

% IVT'
tff(fact_5148_IVT2_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F3: fun(A,B),B2: A,Y: B,A3: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,B2)),Y))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),aa(A,B,F3,A3)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
             => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B2),F3)
               => ? [X3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
                    & ( aa(A,B,F3,X3) = Y ) ) ) ) ) ) ) ).

% IVT2'
tff(fact_5149_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S2: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => ( topolo81223032696312382ous_on(A,B,S2,G3)
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                 => ( aa(A,B,G3,X3) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S2,aa(fun(A,B),fun(A,B),aTP_Lamp_wp(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ) ).

% continuous_on_divide
tff(fact_5150_continuous__on__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [S2: set(A),F3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( aa(A,B,F3,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S2,aTP_Lamp_wq(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_on_inverse
tff(fact_5151_continuous__on__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S2: set(A),F3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( aa(A,B,F3,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S2,aTP_Lamp_wr(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_on_sgn
tff(fact_5152_Bseq__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),G3: fun(nat,A)] :
          ( bfun(nat,A,F3,at_top(nat))
         => ( bfun(nat,A,G3,at_top(nat))
           => bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ws(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G3),at_top(nat)) ) ) ) ).

% Bseq_mult
tff(fact_5153_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_cq(fun(nat,A),fun(nat,A),F3),at_top(nat))
        <=> bfun(nat,A,F3,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_5154_continuous__onI__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & dense_order(B)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( topolo1002775350975398744n_open(B,image2(A,B,F3,A6))
         => ( ! [X3: A,Y3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A6))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) ) ) )
           => topolo81223032696312382ous_on(A,B,A6,F3) ) ) ) ).

% continuous_onI_mono
tff(fact_5155_continuous__on__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S2: set(A),F3: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S2,F3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( cos(A,aa(A,A,F3,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S2,aTP_Lamp_rn(fun(A,A),fun(A,A),F3)) ) ) ) ).

% continuous_on_tan
tff(fact_5156_continuous__on__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S2: set(A),F3: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S2,F3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( sin(A,aa(A,A,F3,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S2,aTP_Lamp_sn(fun(A,A),fun(A,A),F3)) ) ) ) ).

% continuous_on_cot
tff(fact_5157_continuous__on__tanh,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A6: set(C),F3: fun(C,A)] :
          ( topolo81223032696312382ous_on(C,A,A6,F3)
         => ( ! [X3: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),A6))
               => ( cosh(A,aa(C,A,F3,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(C,A,A6,aTP_Lamp_wt(fun(C,A),fun(C,A),F3)) ) ) ) ).

% continuous_on_tanh
tff(fact_5158_Bseq__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cp(A,fun(fun(nat,A),fun(nat,A)),C3),F3),at_top(nat))
          <=> bfun(nat,A,F3,at_top(nat)) ) ) ) ).

% Bseq_cmult_iff
tff(fact_5159_DERIV__atLeastAtMost__imp__continuous__on,axiom,
    ! [A: $tType] :
      ( ( ord(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,B2: A,F3: fun(A,A)] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
               => ? [Y4: A] : has_field_derivative(A,F3,Y4,topolo174197925503356063within(A,X3,top_top(set(A)))) ) )
         => topolo81223032696312382ous_on(A,A,set_or1337092689740270186AtMost(A,A3,B2),F3) ) ) ).

% DERIV_atLeastAtMost_imp_continuous_on
tff(fact_5160_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N7: nat] :
            ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7)))) ) ) ).

% Bseq_iff1a
tff(fact_5161_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N7: nat] :
            ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7)))) ) ) ).

% Bseq_iff
tff(fact_5162_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F3: fun(nat,set(A)),S3: set(A)] :
      ( ! [I3: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F3,I3)),S3))
     => ( pp(aa(set(A),bool,finite_finite2(A),S3))
       => ( ? [N8: nat] :
              ( ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),N8))
                 => ! [M: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N8))
                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N3))
                       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(nat,set(A),F3,M)),aa(nat,set(A),F3,N3))) ) ) )
              & ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N3))
                 => ( aa(nat,set(A),F3,N8) = aa(nat,set(A),F3,N3) ) ) )
         => ( aa(nat,set(A),F3,aa(set(A),nat,finite_card(A),S3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),F3,top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_5163_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),No3))
                  & ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No3),N5))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N5),L5)),R5)) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_5164_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => inj_on(real,real,aTP_Lamp_nh(nat,fun(real,real),N),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_5165_Sup__lessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Y: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_lessThan(A),Y)) = Y ) ).

% Sup_lessThan
tff(fact_5166_Sup__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_atMost(A),Y)) = Y ) ).

% Sup_atMost
tff(fact_5167_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,X,Y)) = Y ) ) ) ).

% Sup_atLeastAtMost
tff(fact_5168_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Y,X)) = X ) ) ) ).

% cSup_atLeastAtMost
tff(fact_5169_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,X,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_5170_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_5171_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,X,Y)) = Y ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_5172_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,X,Y)) = Y ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_5173_Sup__atLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_atLeast(A),X)) = top_top(A) ) ).

% Sup_atLeast
tff(fact_5174_inj__mult__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A] :
          ( inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_mult_left
tff(fact_5175_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),top_top(A)))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),X)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_5176_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( inj_on(A,A,aTP_Lamp_wu(A,fun(A,A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_5177_dist__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: real,A3: A,Y: real] : real_V557655796197034286t_dist(A,aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),X),A3),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),Y),A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y))),real_V7770717601297561774m_norm(A,A3)) ) ).

% dist_scaleR
tff(fact_5178_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X6: set(A),A3: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A3)) )
         => ( ! [Y3: A] :
                ( ! [X5: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),X6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y3)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),Y3)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A3 ) ) ) ) ).

% cSup_eq
tff(fact_5179_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z2: A,X6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),X6))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = Z2 ) ) ) ) ).

% cSup_eq_maximum
tff(fact_5180_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,A6: set(A)] :
          ( ( A3 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),A6) ) ) ).

% inj_on_mult
tff(fact_5181_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S3: set(A),F3: fun(B,A),T5: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S3),image2(B,A,F3,T5)))
    <=> ? [U2: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),U2),T5))
          & inj_on(B,A,F3,U2)
          & ( S3 = image2(B,A,F3,U2) ) ) ) ).

% subset_image_inj
tff(fact_5182_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_5183_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Z2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X6)),Z2)) ) ) ) ).

% cSup_least
tff(fact_5184_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A3: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A3)) )
           => ( ! [Y3: A] :
                  ( ! [X5: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),X6))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y3)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),Y3)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A3 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_5185_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ).

% le_cSup_finite
tff(fact_5186_inj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat] :
      ( inj_on(A,A,F3,top_top(set(A)))
     => inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),top_top(set(A))) ) ).

% inj_fn
tff(fact_5187_inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,B6: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),B6)) = aa(set(A),A,complete_Sup_Sup(A),image2(A,A,aa(A,fun(A,A),inf_inf(A),A3),B6)) ) ).

% inf_Sup
tff(fact_5188_Sup__inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B6: set(A),A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B6)),A3) = bot_bot(A) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B6))
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),A3) = bot_bot(A) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_5189_finite__subset__Union,axiom,
    ! [A: $tType,A6: set(A),B11: set(set(A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B11)))
       => ~ ! [F7: set(set(A))] :
              ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),F7))
             => ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),F7),B11))
               => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F7))) ) ) ) ) ).

% finite_subset_Union
tff(fact_5190_card__Union__le__sum__card,axiom,
    ! [A: $tType,U3: set(set(A))] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),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_5191_SUP__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F3: fun(B,A),B6: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,B6))),A3) = aa(set(A),A,complete_Sup_Sup(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_wv(fun(B,A),fun(A,fun(B,A)),F3),A3),B6)) ) ).

% SUP_inf
tff(fact_5192_Sup__inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B6: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B6)),A3) = aa(set(A),A,complete_Sup_Sup(A),image2(A,A,aTP_Lamp_ww(A,fun(A,A),A3),B6)) ) ).

% Sup_inf
tff(fact_5193_inf__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F3: fun(B,A),B6: set(B)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,B6))) = aa(set(A),A,complete_Sup_Sup(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_wx(A,fun(fun(B,A),fun(B,A)),A3),F3),B6)) ) ).

% inf_SUP
tff(fact_5194_SUP__inf__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F3: fun(B,A),A6: set(B),G3: fun(C,A),B6: set(C)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Sup_Sup(A),image2(C,A,G3,B6))) = aa(set(A),A,complete_Sup_Sup(A),image2(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_wz(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F3),G3),B6),A6)) ) ).

% SUP_inf_distrib2
tff(fact_5195_UN__lessThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),image2(nat,set(nat),set_ord_lessThan(nat),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_lessThan_UNIV
tff(fact_5196_UN__atMost__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),image2(nat,set(nat),set_ord_atMost(nat),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atMost_UNIV
tff(fact_5197_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A6: set(A),A10: set(B)] :
      ( ( A6 != bot_bot(set(A)) )
     => ( ? [F5: fun(A,B)] :
            ( inj_on(A,B,F5,A6)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image2(A,B,F5,A6)),A10)) )
      <=> ? [G6: fun(B,A)] : image2(B,A,G6,A10) = A6 ) ) ).

% inj_on_iff_surj
tff(fact_5198_cSUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),F3: fun(B,A),M5: A] :
          ( ( A6 != bot_bot(set(B)) )
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),M5)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),M5)) ) ) ) ).

% cSUP_least
tff(fact_5199_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),A3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S3))),A3)) ) ) ) ).

% cSup_abs_le
tff(fact_5200_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U3: set(set(A))] :
      ( ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),U3))
         => pp(aa(set(A),bool,finite_finite2(A),X3)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),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_5201_UN__atLeast__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),image2(nat,set(nat),set_ord_atLeast(nat),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atLeast_UNIV
tff(fact_5202_UN__UN__finite__eq,axiom,
    ! [A: $tType,A6: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),aTP_Lamp_xa(fun(nat,set(A)),fun(nat,set(A)),A6),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),A6,top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_5203_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M5: fun(nat,set(A)),K2: nat,N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_xb(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M5),K2),aa(nat,set(nat),set_ord_lessThan(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),M5,set_or7035219750837199246ssThan(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2)))) ).

% UN_le_add_shift_strict
tff(fact_5204_UN__le__add__shift,axiom,
    ! [A: $tType,M5: fun(nat,set(A)),K2: nat,N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_xb(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M5),K2),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),M5,set_or1337092689740270186AtMost(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2)))) ).

% UN_le_add_shift
tff(fact_5205_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),L: A,E3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),L))),E3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Sup_Sup(A),S3)),L))),E3)) ) ) ) ).

% cSup_asclose
tff(fact_5206_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
             => ? [M10: nat] :
                ! [M: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),M))
                 => ! [N3: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),N3))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M),aa(nat,A,X6,N3))),E2)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% metric_CauchyI
tff(fact_5207_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),E3: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
           => ? [M8: nat] :
              ! [M3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M3))
               => ! [N4: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),N4))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M3),aa(nat,A,X6,N4))),E3)) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_5208_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S2)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [N7: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,S2,N5),aa(nat,A,S2,N7))),E4)) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_5209_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [M9: nat] :
                ! [M6: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M6))
                 => ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),N5))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M6),aa(nat,A,X6,N5))),E4)) ) ) ) ) ) ).

% Cauchy_def
tff(fact_5210_Sup__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S2: set(A),A3: A] :
          ( ! [N3: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,B2,N3)),S2))
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(set(A),A,complete_Sup_Sup(A),S2))) ) ) ) ).

% Sup_lim
tff(fact_5211_UN__finite__subset,axiom,
    ! [A: $tType,A6: fun(nat,set(A)),C5: set(A)] :
      ( ! [N3: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),A6,set_or7035219750837199246ssThan(nat,zero_zero(nat),N3)))),C5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),A6,top_top(set(nat))))),C5)) ) ).

% UN_finite_subset
tff(fact_5212_UN__finite2__eq,axiom,
    ! [A: $tType,A6: fun(nat,set(A)),B6: fun(nat,set(A)),K2: nat] :
      ( ! [N3: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),A6,set_or7035219750837199246ssThan(nat,zero_zero(nat),N3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),B6,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),K2))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),A6,top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),B6,top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_5213_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),I5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),image2(A,set(B),A6,I5)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_xc(fun(A,set(B)),fun(A,nat),A6)),I5))) ) ).

% card_UN_le
tff(fact_5214_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B6))
       => ( ? [F5: fun(A,B)] :
              ( inj_on(A,B,F5,A6)
              & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image2(A,B,F5,A6)),B6)) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B6))) ) ) ) ).

% inj_on_iff_card_le
tff(fact_5215_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(A),B6: set(B)] :
      ( inj_on(A,B,F3,A6)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image2(A,B,F3,A6)),B6))
       => ( pp(aa(set(B),bool,finite_finite2(B),B6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B6))) ) ) ) ).

% card_inj_on_le
tff(fact_5216_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B6))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B6)))
         => ? [F2: fun(A,B)] :
              ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image2(A,B,F2,A6)),B6))
              & inj_on(A,B,F2,A6) ) ) ) ) ).

% card_le_inj
tff(fact_5217_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F3: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F3)
        <=> ! [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
             => ? [M9: nat] :
                ! [M6: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M6))
                 => ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M6),N5))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F3,M6),aa(nat,A,F3,N5))),E4)) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_5218_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
             => ? [M10: nat] :
                ! [M: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),M))
                 => ! [N3: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N3))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M),aa(nat,A,X6,N3))),E2)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI'
tff(fact_5219_card__partition,axiom,
    ! [A: $tType,C5: set(set(A)),K2: nat] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),C5))
     => ( pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5)))
       => ( ! [C2: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C2),C5))
             => ( aa(set(A),nat,finite_card(A),C2) = K2 ) )
         => ( ! [C1: set(A),C22: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C1),C5))
               => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C22),C5))
                 => ( ( C1 != C22 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C1),C22) = bot_bot(set(A)) ) ) ) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(set(set(A)),nat,finite_card(set(A)),C5)) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5)) ) ) ) ) ) ).

% card_partition
tff(fact_5220_UN__finite2__subset,axiom,
    ! [A: $tType,A6: fun(nat,set(A)),B6: fun(nat,set(A)),K2: nat] :
      ( ! [N3: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),A6,set_or7035219750837199246ssThan(nat,zero_zero(nat),N3)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),B6,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),K2))))))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),A6,top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),B6,top_top(set(nat)))))) ) ).

% UN_finite2_subset
tff(fact_5221_UN__le__eq__Un0,axiom,
    ! [A: $tType,M5: fun(nat,set(A)),N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),M5,aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),M5,set_or1337092689740270186AtMost(nat,one_one(nat),N)))),aa(nat,set(A),M5,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_5222_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A,R: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
           => ? [No: nat] :
              ! [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No),N4))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N4),L5)),R)) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_5223_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
             => ? [No2: nat] :
                ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No2),N3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N3),L5)),R3)) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_5224_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No3: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No3),N5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N5),L5)),R5)) ) ) ) ) ).

% lim_sequentially
tff(fact_5225_map__sorted__distinct__set__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),Ys: list(B)] :
          ( inj_on(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)))
         => ( sorted_wrt(A,ord_less_eq(A),map(B,A,F3,Xs))
           => ( distinct(A,map(B,A,F3,Xs))
             => ( sorted_wrt(A,ord_less_eq(A),map(B,A,F3,Ys))
               => ( distinct(A,map(B,A,F3,Ys))
                 => ( ( aa(list(B),set(B),set2(B),Xs) = aa(list(B),set(B),set2(B),Ys) )
                   => ( Xs = Ys ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_5226_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [J: nat] :
            ? [M9: nat] :
            ! [M6: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M6))
             => ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),N5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M6),aa(nat,A,X6,N5))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J))))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_5227_map__of__mapk__SomeI,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,B),T2: list(product_prod(A,C)),K2: A,X: C] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( ( aa(A,option(C),map_of(A,C,T2),K2) = aa(C,option(C),some(C),X) )
       => ( aa(B,option(C),map_of(B,C,map(product_prod(A,C),product_prod(B,C),aa(fun(A,fun(C,product_prod(B,C))),fun(product_prod(A,C),product_prod(B,C)),product_case_prod(A,C,product_prod(B,C)),aTP_Lamp_xd(fun(A,B),fun(A,fun(C,product_prod(B,C))),F3)),T2)),aa(A,B,F3,K2)) = aa(C,option(C),some(C),X) ) ) ) ).

% map_of_mapk_SomeI
tff(fact_5228_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: A,B6: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A6),aa(set(A),A,complete_Sup_Sup(A),image2(nat,A,aTP_Lamp_xe(A,fun(nat,A),B6),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A6),B6) ) ).

% SUP_nat_binary
tff(fact_5229_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image2(A,B,F3,A6))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),A6)))) ) ) ).

% mono_Sup
tff(fact_5230_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A6: fun(C,A),I5: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xf(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A6),I5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),image2(C,A,A6,I5))))) ) ) ).

% mono_SUP
tff(fact_5231_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_5232_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(set(product_prod(A,B))),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),image2(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),S3)),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S3))) ) ).

% SUP_Sup_eq2
tff(fact_5233_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: fun(C,set(product_prod(A,B))),S3: set(C),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),aTP_Lamp_xg(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R),S3)),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R,S3)))) ) ).

% SUP_UN_eq2
tff(fact_5234_inj__list__encode,axiom,
    ! [A6: set(list(nat))] : inj_on(list(nat),nat,nat_list_encode,A6) ).

% inj_list_encode
tff(fact_5235_inj__list__decode,axiom,
    ! [A6: set(nat)] : inj_on(nat,list(nat),nat_list_decode,A6) ).

% inj_list_decode
tff(fact_5236_inj__prod__encode,axiom,
    ! [A6: set(product_prod(nat,nat))] : inj_on(product_prod(nat,nat),nat,nat_prod_encode,A6) ).

% inj_prod_encode
tff(fact_5237_inj__on__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N6: set(nat)] : inj_on(nat,A,semiring_1_of_nat(A),N6) ) ).

% inj_on_of_nat
tff(fact_5238_inj__Suc,axiom,
    ! [N6: set(nat)] : inj_on(nat,nat,suc,N6) ).

% inj_Suc
tff(fact_5239_inj__Some,axiom,
    ! [A: $tType,A6: set(A)] : inj_on(A,option(A),some(A),A6) ).

% inj_Some
tff(fact_5240_inj__prod__decode,axiom,
    ! [A6: set(nat)] : inj_on(nat,product_prod(nat,nat),nat_prod_decode,A6) ).

% inj_prod_decode
tff(fact_5241_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X6: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_xh(fun(A,B),fun(A,product_prod(A,B)),F3),X6) ).

% inj_on_convol_ident
tff(fact_5242_inj__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => inj_on(nat,A,semiring_1_of_nat(A),top_top(set(nat))) ) ).

% inj_of_nat
tff(fact_5243_inj__on__diff__nat,axiom,
    ! [N6: set(nat),K2: nat] :
      ( ! [N3: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N3),N6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N3)) )
     => inj_on(nat,nat,aTP_Lamp_kt(nat,fun(nat,nat),K2),N6) ) ).

% inj_on_diff_nat
tff(fact_5244_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(fun(A,fun(B,bool))),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),S3),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image2(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),image2(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),S3))))) ) ).

% Sup_SUP_eq2
tff(fact_5245_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A6: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_xi(A,fun(B,product_prod(B,A)))),A6) ).

% swap_inj_on
tff(fact_5246_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,aa(fun(set(nat),bool),set(set(nat)),collect(set(nat)),finite_finite2(nat))) ).

% inj_on_set_encode
tff(fact_5247_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X3: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),I5))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs))) )
       => inj_on(nat,A,nth(A,Xs),I5) ) ) ).

% inj_on_nth
tff(fact_5248_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S3))
    <=> ? [F5: fun(nat,A)] :
          ( inj_on(nat,A,F5,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image2(nat,A,F5,top_top(set(nat)))),S3)) ) ) ).

% infinite_iff_countable_subset
tff(fact_5249_infinite__countable__subset,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),S3))
     => ? [F2: fun(nat,A)] :
          ( inj_on(nat,A,F2,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image2(nat,A,F2,top_top(set(nat)))),S3)) ) ) ).

% infinite_countable_subset
tff(fact_5250_inj__on__funpow__least,axiom,
    ! [A: $tType,N: nat,F3: fun(A,A),S2: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),S2) = S2 )
     => ( ! [M: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3),S2) != S2 ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_xj(fun(A,A),fun(A,fun(nat,A)),F3),S2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).

% inj_on_funpow_least
tff(fact_5251_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),X: A] :
          ( ! [Y3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
         => ( ! [Y3: A] :
                ( ! [Z4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),Y3)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A6) = X ) ) ) ) ).

% Sup_eqI
tff(fact_5252_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B6: set(A)] :
          ( ! [A5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A6))
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),B6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X5)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B6))) ) ) ).

% Sup_mono
tff(fact_5253_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),Z2: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),Z2)) ) ) ).

% Sup_least
tff(fact_5254_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ).

% Sup_upper
tff(fact_5255_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),B2))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2)) ) ) ) ).

% Sup_le_iff
tff(fact_5256_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A6: set(A),V3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V3),U))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V3),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ) ).

% Sup_upper2
tff(fact_5257_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,A6: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A6)))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X4)) ) ) ) ) ).

% le_Sup_iff
tff(fact_5258_SUP__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),B6: set(C),F3: fun(B,A),G3: fun(C,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X5),B6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),aa(C,A,G3,X5))) ) )
         => ( ! [J3: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),J3),B6))
               => ? [X5: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A6))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G3,J3)),aa(B,A,F3,X5))) ) )
           => ( aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6)) = aa(set(A),A,complete_Sup_Sup(A),image2(C,A,G3,B6)) ) ) ) ) ).

% SUP_eq
tff(fact_5259_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),U: A] :
          ( ! [V2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V2)) )
         => ( ( A6 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ) ).

% less_eq_Sup
tff(fact_5260_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B6: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B6))) ) ) ).

% Sup_subset_mono
tff(fact_5261_SUP__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),F3: fun(B,A),X: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),X)) )
         => ( ! [Y3: A] :
                ( ! [I4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I4)),Y3)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6)) = X ) ) ) ) ).

% SUP_eqI
tff(fact_5262_SUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),B6: set(C),F3: fun(B,A),G3: fun(C,A)] :
          ( ! [N3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),N3),A6))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X5),B6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,N3)),aa(C,A,G3,X5))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Sup_Sup(A),image2(C,A,G3,B6)))) ) ) ).

% SUP_mono
tff(fact_5263_SUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),F3: fun(B,A),U: A] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),U)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),U)) ) ) ).

% SUP_least
tff(fact_5264_SUP__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A6: set(B)] :
          ( ! [X3: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,G3,A6)))) ) ) ).

% SUP_mono'
tff(fact_5265_SUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6)))) ) ) ).

% SUP_upper
tff(fact_5266_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A6: set(B),U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),U))
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X4)),U)) ) ) ) ).

% SUP_le_iff
tff(fact_5267_SUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A6: set(B),U: A,F3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,I2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6)))) ) ) ) ).

% SUP_upper2
tff(fact_5268_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,F3: fun(B,A),A6: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X))
             => ? [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),aa(B,A,F3,X4))) ) ) ) ) ).

% le_SUP_iff
tff(fact_5269_SUP__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),C3: A,F3: fun(B,A)] :
          ( ( I5 != bot_bot(set(B)) )
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(B,A,F3,I3))) )
           => ( ( aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,I5)) = C3 )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),I5))
                 => ( aa(B,A,F3,X4) = C3 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_5270_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B6: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B6)))) ) ).

% Sup_inter_less_eq
tff(fact_5271_SUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),B6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,G3,B6)))) ) ) ) ).

% SUP_subset_mono
tff(fact_5272_card__UNION,axiom,
    ! [A: $tType,A6: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),A6))
     => ( ! [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),A6))
           => pp(aa(set(A),bool,finite_finite2(A),X3)) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A6)) = 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_xk(set(set(A)),int)),aa(fun(set(set(A)),bool),set(set(set(A))),collect(set(set(A))),aTP_Lamp_xl(set(set(A)),fun(set(set(A)),bool),A6)))) ) ) ) ).

% card_UNION
tff(fact_5273_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A6: set(A),F3: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A6))
                 => ( aa(A,B,F3,X3) != aa(A,B,F3,Y3) ) ) ) )
         => ( ! [X3: A,Y3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A6))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
                    | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X3)) ) ) )
           => inj_on(A,B,F3,A6) ) ) ) ).

% linorder_inj_onI
tff(fact_5274_length__remdups__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),remdups(A,concat(A,Xss))) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(list(A),set(A),set2(A),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))) ).

% length_remdups_concat
tff(fact_5275_Inf__greaterThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_greaterThan(A),X)) = X ) ).

% Inf_greaterThan
tff(fact_5276_Inf__atLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_atLeast(A),X)) = X ) ).

% Inf_atLeast
tff(fact_5277_length__remdups__eq,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),remdups(A,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ( remdups(A,Xs) = Xs ) ) ).

% length_remdups_eq
tff(fact_5278_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,X,Y)) = X ) ) ) ).

% Inf_atLeastAtMost
tff(fact_5279_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Y,X)) = Y ) ) ) ).

% cInf_atLeastAtMost
tff(fact_5280_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,X,Y)) = X ) ) ) ).

% Inf_atLeastLessThan
tff(fact_5281_Inf__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_atMost(A),X)) = bot_bot(A) ) ).

% Inf_atMost
tff(fact_5282_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,X,Y)) = X ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_5283_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,X,Y)) = X ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_5284_length__remdups__leq,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_remdups_leq
tff(fact_5285_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_lessThan(A),X)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_5286_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z2: A,X6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),X6))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X3)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = Z2 ) ) ) ) ).

% cInf_eq_minimum
tff(fact_5287_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X6: set(A),A3: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3)) )
         => ( ! [Y3: A] :
                ( ! [X5: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),X6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X5)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),A3)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A3 ) ) ) ) ).

% cInf_eq
tff(fact_5288_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),X: A] :
          ( ! [I3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),I3)) )
         => ( ! [Y3: A] :
                ( ! [I4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),I4)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A6) = X ) ) ) ) ).

% Inf_eqI
tff(fact_5289_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B6: set(A),A6: set(A)] :
          ( ! [B4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B6))
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),B4)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B6))) ) ) ).

% Inf_mono
tff(fact_5290_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),X)) ) ) ).

% Inf_lower
tff(fact_5291_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A6: set(A),V3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),V3)) ) ) ) ).

% Inf_lower2
tff(fact_5292_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A6: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A6)))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X4)) ) ) ) ).

% le_Inf_iff
tff(fact_5293_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),Z2: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(set(A),A,complete_Inf_Inf(A),A6))) ) ) ).

% Inf_greatest
tff(fact_5294_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),X))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y5))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5)) ) ) ) ) ).

% Inf_le_iff
tff(fact_5295_INF__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),B6: set(C),G3: fun(C,A),F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X5),B6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G3,X5)),aa(B,A,F3,I3))) ) )
         => ( ! [J3: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),J3),B6))
               => ? [X5: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A6))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X5)),aa(C,A,G3,J3))) ) )
           => ( aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6)) = aa(set(A),A,complete_Inf_Inf(A),image2(C,A,G3,B6)) ) ) ) ) ).

% INF_eq
tff(fact_5296_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Z2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(set(A),A,complete_Inf_Inf(A),X6))) ) ) ) ).

% cInf_greatest
tff(fact_5297_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A3: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3)) )
           => ( ! [Y3: A] :
                  ( ! [X5: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),X6))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X5)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),A3)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A3 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_5298_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),U: A] :
          ( ! [V2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V2),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V2),U)) )
         => ( ( A6 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),U)) ) ) ) ).

% Inf_less_eq
tff(fact_5299_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),X6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),X)) ) ) ) ).

% cInf_le_finite
tff(fact_5300_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B6: set(A),A6: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B6))) ) ) ).

% Inf_superset_mono
tff(fact_5301_sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,B6: set(A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),B6)) = aa(set(A),A,complete_Inf_Inf(A),image2(A,A,aa(A,fun(A,A),sup_sup(A),A3),B6)) ) ).

% sup_Inf
tff(fact_5302_Inf__sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B6: set(A),A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B6)),A3) = top_top(A) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B6))
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),A3) = top_top(A) ) ) ) ) ).

% Inf_sup_eq_top_iff
tff(fact_5303_INF__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),X: A,F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(B,A,F3,I3))) )
         => ( ! [Y3: A] :
                ( ! [I4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),aa(B,A,F3,I4))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6)) = X ) ) ) ) ).

% INF_eqI
tff(fact_5304_INF__mono,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B6: set(B),A6: set(C),F3: fun(C,A),G3: fun(B,A)] :
          ( ! [M: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),M),B6))
             => ? [X5: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X5),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F3,X5)),aa(B,A,G3,M))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(C,A,F3,A6))),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,G3,B6)))) ) ) ).

% INF_mono
tff(fact_5305_INF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),aa(B,A,F3,I2))) ) ) ).

% INF_lower
tff(fact_5306_INF__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A6: set(B)] :
          ( ! [X3: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,G3,A6)))) ) ) ).

% INF_mono'
tff(fact_5307_INF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A6: set(B),F3: fun(B,A),U: A] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),U))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),U)) ) ) ) ).

% INF_lower2
tff(fact_5308_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F3: fun(B,A),A6: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))))
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,X4))) ) ) ) ).

% le_INF_iff
tff(fact_5309_INF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),U: A,F3: fun(B,A)] :
          ( ! [I3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,I3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6)))) ) ) ).

% INF_greatest
tff(fact_5310_INF__sup__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F3: fun(B,A),A6: set(B),G3: fun(C,A),B6: set(C)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Inf_Inf(A),image2(C,A,G3,B6))) = aa(set(A),A,complete_Inf_Inf(A),image2(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_xn(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F3),G3),B6),A6)) ) ).

% INF_sup_distrib2
tff(fact_5311_sup__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F3: fun(B,A),B6: set(B)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,B6))) = aa(set(A),A,complete_Inf_Inf(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xo(A,fun(fun(B,A),fun(B,A)),A3),F3),B6)) ) ).

% sup_INF
tff(fact_5312_Inf__sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B6: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B6)),A3) = aa(set(A),A,complete_Inf_Inf(A),image2(A,A,aTP_Lamp_xp(A,fun(A,A),A3),B6)) ) ).

% Inf_sup
tff(fact_5313_INF__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F3: fun(B,A),B6: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,B6))),A3) = aa(set(A),A,complete_Inf_Inf(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_xq(fun(B,A),fun(A,fun(B,A)),F3),A3),B6)) ) ).

% INF_sup
tff(fact_5314_sorted__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remdups(A,Xs)) ) ) ).

% sorted_remdups
tff(fact_5315_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A6: set(B),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),X))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y5))
             => ? [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X4)),Y5)) ) ) ) ) ).

% INF_le_iff
tff(fact_5316_cINF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),M2: A,F3: fun(B,A)] :
          ( ( A6 != bot_bot(set(B)) )
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),aa(B,A,F3,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6)))) ) ) ) ).

% cINF_greatest
tff(fact_5317_INF__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),F3: fun(B,A),C3: A] :
          ( ( I5 != bot_bot(set(B)) )
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),C3)) )
           => ( ( aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,I5)) = C3 )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),I5))
                 => ( aa(B,A,F3,X4) = C3 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_5318_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A)] :
          ( ( A6 != bot_bot(set(A)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ).

% Inf_le_Sup
tff(fact_5319_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),A3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S3))),A3)) ) ) ) ).

% cInf_abs_ge
tff(fact_5320_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A),B6: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6)))) ) ).

% less_eq_Inf_inter
tff(fact_5321_INF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B6: set(B),A6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),A6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),B6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,G3,B6)))) ) ) ) ).

% INF_superset_mono
tff(fact_5322_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A6) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_xr(set(A),fun(A,bool),A6))) ) ).

% Inf_eq_Sup
tff(fact_5323_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A6) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_xs(set(A),fun(A,bool),A6))) ) ).

% Sup_eq_Inf
tff(fact_5324_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A6))),aa(set(B),B,complete_Inf_Inf(B),image2(A,B,F3,A6)))) ) ) ).

% mono_Inf
tff(fact_5325_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F3: fun(A,B),A6: fun(C,A),I5: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),image2(C,A,A6,I5)))),aa(set(B),B,complete_Inf_Inf(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xf(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A6),I5)))) ) ) ).

% mono_INF
tff(fact_5326_length__remdups__card__conv,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),remdups(A,Xs)) = aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) ).

% length_remdups_card_conv
tff(fact_5327_INF__le__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(B),F3: fun(B,A)] :
          ( ( A6 != bot_bot(set(B)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6)))) ) ) ).

% INF_le_SUP
tff(fact_5328_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),L: A,E3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),L))),E3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Inf_Inf(A),S3)),L))),E3)) ) ) ) ).

% cInf_asclose
tff(fact_5329_Inf__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S2: set(A),A3: A] :
          ( ! [N3: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,B2,N3)),S2))
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),S2)),A3)) ) ) ) ).

% Inf_lim
tff(fact_5330_mono__bij__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & comple5582772986160207858norder(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( bij_betw(A,B,F3,top_top(set(A)),top_top(set(B)))
           => ( aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A6)) = aa(set(B),B,complete_Inf_Inf(B),image2(A,B,F3,A6)) ) ) ) ) ).

% mono_bij_Inf
tff(fact_5331_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: A,B6: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A6),aa(set(A),A,complete_Inf_Inf(A),image2(nat,A,aTP_Lamp_xe(A,fun(nat,A),B6),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A6),B6) ) ).

% INF_nat_binary
tff(fact_5332_INF__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P2: fun(C,fun(B,A))] : aa(set(A),A,complete_Inf_Inf(A),image2(B,A,aTP_Lamp_xu(fun(C,fun(B,A)),fun(B,A),P2),top_top(set(B)))) = aa(set(A),A,complete_Sup_Sup(A),image2(fun(B,C),A,aTP_Lamp_xw(fun(C,fun(B,A)),fun(fun(B,C),A),P2),top_top(set(fun(B,C))))) ) ).

% INF_SUP
tff(fact_5333_SUP__INF,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P2: fun(C,fun(B,A))] : aa(set(A),A,complete_Sup_Sup(A),image2(B,A,aTP_Lamp_xx(fun(C,fun(B,A)),fun(B,A),P2),top_top(set(B)))) = aa(set(A),A,complete_Inf_Inf(A),image2(fun(B,C),A,aTP_Lamp_xy(fun(C,fun(B,A)),fun(fun(B,C),A),P2),top_top(set(fun(B,C))))) ) ).

% SUP_INF
tff(fact_5334_INT__greaterThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),image2(nat,set(nat),set_ord_greaterThan(nat),top_top(set(nat)))) = bot_bot(set(nat)) ).

% INT_greaterThan_UNIV
tff(fact_5335_inj__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] :
      ( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3),top_top(set(product_prod(A,B))))
    <=> inj_on(B,C,F3,top_top(set(B))) ) ).

% inj_apsnd
tff(fact_5336_inj__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F3),top_top(set(product_prod(A,B))))
    <=> inj_on(A,C,F3,top_top(set(A))) ) ).

% inj_apfst
tff(fact_5337_Gcd__eq__Max,axiom,
    ! [M5: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),M5))
     => ( ( M5 != bot_bot(set(nat)) )
       => ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M5))
         => ( gcd_Gcd(nat,M5) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),image2(nat,set(nat),aTP_Lamp_xz(nat,set(nat)),M5))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_5338_apfst__conv,axiom,
    ! [C: $tType,A: $tType,B: $tType,F3: fun(C,A),X: C,Y: B] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,X)),Y) ).

% apfst_conv
tff(fact_5339_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,B),X: A,Y: C] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),aa(C,B,F3,Y)) ).

% apsnd_conv
tff(fact_5340_apfst__eq__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),X: product_prod(C,B),G3: fun(C,A)] :
      ( ( aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),X) = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,G3),X) )
    <=> ( aa(C,A,F3,aa(product_prod(C,B),C,product_fst(C,B),X)) = aa(C,A,G3,aa(product_prod(C,B),C,product_fst(C,B),X)) ) ) ).

% apfst_eq_conv
tff(fact_5341_fst__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),X: product_prod(C,B)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),X)) = aa(C,A,F3,aa(product_prod(C,B),C,product_fst(C,B),X)) ).

% fst_apfst
tff(fact_5342_snd__apfst,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(C,B),X: product_prod(C,A)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,A),product_prod(B,A),product_apfst(C,B,A,F3),X)) = aa(product_prod(C,A),A,product_snd(C,A),X) ).

% snd_apfst
tff(fact_5343_fst__apsnd,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),X: product_prod(A,C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),X)) = aa(product_prod(A,C),A,product_fst(A,C),X) ).

% fst_apsnd
tff(fact_5344_apsnd__eq__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),X: product_prod(A,C),G3: fun(C,B)] :
      ( ( aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),X) = aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),G3),X) )
    <=> ( aa(C,B,F3,aa(product_prod(A,C),C,product_snd(A,C),X)) = aa(C,B,G3,aa(product_prod(A,C),C,product_snd(A,C),X)) ) ) ).

% apsnd_eq_conv
tff(fact_5345_snd__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),X: product_prod(B,C)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(B,C),product_prod(B,A),aa(fun(C,A),fun(product_prod(B,C),product_prod(B,A)),product_apsnd(C,A,B),F3),X)) = aa(C,A,F3,aa(product_prod(B,C),C,product_snd(B,C),X)) ).

% snd_apsnd
tff(fact_5346_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_bo(nat,fun(nat,bool),N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_5347_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A6)),X))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_5348_apsnd__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: fun(C,B),G3: fun(D,A),X: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G3),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G3,aa(product_prod(D,C),D,product_fst(D,C),X))),aa(C,B,F3,aa(product_prod(D,C),C,product_snd(D,C),X))) ).

% apsnd_apfst
tff(fact_5349_apfst__apsnd,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,F3: fun(C,A),G3: fun(D,B),X: product_prod(C,D)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(product_prod(C,D),product_prod(C,B),aa(fun(D,B),fun(product_prod(C,D),product_prod(C,B)),product_apsnd(D,B,C),G3),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,aa(product_prod(C,D),C,product_fst(C,D),X))),aa(D,B,G3,aa(product_prod(C,D),D,product_snd(C,D),X))) ).

% apfst_apsnd
tff(fact_5350_apsnd__apfst__commute,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: fun(C,B),G3: fun(D,A),P: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G3),P)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,G3),aa(product_prod(D,C),product_prod(D,B),aa(fun(C,B),fun(product_prod(D,C),product_prod(D,B)),product_apsnd(C,B,D),F3),P)) ).

% apsnd_apfst_commute
tff(fact_5351_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(set(A),A,lattic643756798349783984er_Max(A),A6))) ) ) ) ).

% Max.coboundedI
tff(fact_5352_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),B6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,finite_finite2(A),B6))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
                 => ? [Xa: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),B6))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa)) ) )
             => ( ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B6))
                   => ? [Xa: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),A6))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa)) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A6) = aa(set(A),A,lattic643756798349783984er_Max(A),B6) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_5353_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ! [Y3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A6) = X ) ) ) ) ) ).

% Max_eqI
tff(fact_5354_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A6))) ) ) ) ).

% Max_ge
tff(fact_5355_enumerate__Suc__eq,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : enumerate(A,aa(nat,nat,suc,N),Xs) = map(product_prod(nat,A),product_prod(nat,A),product_apfst(nat,nat,A,suc),enumerate(A,N,Xs)) ).

% enumerate_Suc_eq
tff(fact_5356_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A6)),X)) ) ) ) ) ).

% Max.boundedI
tff(fact_5357_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A6)),X))
             => ! [A8: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A8),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A8),X)) ) ) ) ) ) ).

% Max.boundedE
tff(fact_5358_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),M2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ( M2 = aa(set(A),A,lattic643756798349783984er_Max(A),A6) )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M2),A6))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M2)) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_5359_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A6)))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_5360_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),M2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A6) = M2 )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M2),A6))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M2)) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_5361_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ! [B4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B4),A3)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,A3),A6)) = A3 ) ) ) ) ).

% Max_insert2
tff(fact_5362_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(set(product_prod(A,B))),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),image2(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),S3)),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),S3))) ) ).

% INF_Int_eq2
tff(fact_5363_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: fun(C,set(product_prod(A,B))),S3: set(C),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),aTP_Lamp_xg(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R),S3)),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R,S3)))) ) ).

% INF_INT_eq2
tff(fact_5364_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A),B6: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A6)),aa(set(A),A,lattic643756798349783984er_Max(A),B6))) ) ) ) ) ).

% Max.subset_imp
tff(fact_5365_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M5: set(A),N6: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M5),N6))
         => ( ( M5 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),N6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M5)),aa(set(A),A,lattic643756798349783984er_Max(A),N6))) ) ) ) ) ).

% Max_mono
tff(fact_5366_Sup__nat__def,axiom,
    ! [X6: set(nat)] :
      ( ( ( X6 = bot_bot(set(nat)) )
       => ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = zero_zero(nat) ) )
      & ( ( X6 != bot_bot(set(nat)) )
       => ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),X6) ) ) ) ).

% Sup_nat_def
tff(fact_5367_card__le__Suc__Max,axiom,
    ! [S3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),S3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S3)),aa(nat,nat,suc,aa(set(nat),nat,lattic643756798349783984er_Max(nat),S3)))) ) ).

% card_le_Suc_Max
tff(fact_5368_divide__nat__def,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( divide_divide(nat,M2,N) = zero_zero(nat) ) )
      & ( ( N != zero_zero(nat) )
       => ( divide_divide(nat,M2,N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ya(nat,fun(nat,fun(nat,bool)),N),M2))) ) ) ) ).

% divide_nat_def
tff(fact_5369_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(fun(A,fun(B,bool))),X5: A,Xa: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),S3),X5),Xa))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),image2(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),image2(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),S3))))) ) ).

% Inf_INT_eq2
tff(fact_5370_sum__le__card__Max,axiom,
    ! [A: $tType,A6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A6)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),image2(A,nat,F3,A6))))) ) ).

% sum_le_card_Max
tff(fact_5371_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_VEBT_valid(X,Xa2)
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y)
          <=> ( Xa2 != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
             => ( pp(Y)
              <=> ~ ( ( Deg2 = Xa2 )
                    & ! [X4: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => vEBT_VEBT_valid(X4,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
                    & vEBT_VEBT_valid(Summary2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
                    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                    & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_yb(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_5372_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_VEBT_valid(X,Xa2)
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X5: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                     => vEBT_VEBT_valid(X5,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
                  & vEBT_VEBT_valid(Summary2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
                  & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                  & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_yb(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_5373_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_VEBT_valid(X,Xa2)
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa2 = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
             => ( ( Deg2 = Xa2 )
                & ! [X3: vEBT_VEBT] :
                    ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                   => vEBT_VEBT_valid(X3,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
                & vEBT_VEBT_valid(Summary2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
                & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_yb(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_5374_open__diagonal__complement,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_yc(product_prod(A,A),bool))) ) ).

% open_diagonal_complement
tff(fact_5375_option_Osimps_I4_J,axiom,
    ! [A: $tType,B: $tType,F1: B,F22: fun(A,B)] : aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),none(A)) = F1 ).

% option.simps(4)
tff(fact_5376_option_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F22: fun(A,B),Option: option(A)] : aa(B,C,H,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option)) = aa(option(A),C,aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),case_option(C,A),aa(B,C,H,F1)),aa(fun(A,B),fun(A,C),aTP_Lamp_yd(fun(B,C),fun(fun(A,B),fun(A,C)),H),F22)),Option) ).

% option.case_distrib
tff(fact_5377_option_Osimps_I5_J,axiom,
    ! [B: $tType,A: $tType,F1: B,F22: fun(A,B),X2: A] : aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),aa(A,option(A),some(A),X2)) = aa(A,B,F22,X2) ).

% option.simps(5)
tff(fact_5378_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),fTrue),aTP_Lamp_ye(A,bool)),Option)) ) ).

% option.disc_eq_case(1)
tff(fact_5379_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),fFalse),aTP_Lamp_yf(A,bool)),Option)) ) ).

% option.disc_eq_case(2)
tff(fact_5380_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_yg(product_prod(A,A),bool))) ) ).

% open_subdiagonal
tff(fact_5381_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_yh(product_prod(A,A),bool))) ) ).

% open_superdiagonal
tff(fact_5382_listrel1__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : listrel1(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_yi(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).

% listrel1_def
tff(fact_5383_option_Ocase__eq__if,axiom,
    ! [B: $tType,A: $tType,Option: option(A),F1: B,F22: fun(A,B)] :
      ( ( ( Option = none(A) )
       => ( aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option) = F1 ) )
      & ( ( Option != none(A) )
       => ( aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option) = aa(A,B,F22,aa(option(A),A,the2(A),Option)) ) ) ) ).

% option.case_eq_if
tff(fact_5384_case__optionE,axiom,
    ! [A: $tType,P2: bool,Q: fun(A,bool),X: option(A)] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),P2),Q),X))
     => ( ( ( X = none(A) )
         => ~ pp(P2) )
       => ~ ! [Y3: A] :
              ( ( X = aa(A,option(A),some(A),Y3) )
             => ~ pp(aa(A,bool,Q,Y3)) ) ) ) ).

% case_optionE
tff(fact_5385_lex__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lex(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_yj(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).

% lex_conv
tff(fact_5386_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_yk(list(A),fun(A,bool),Xs)) ).

% set_conv_nth
tff(fact_5387_option_Osplit__sel__asm,axiom,
    ! [B: $tType,A: $tType,P2: fun(B,bool),F1: B,F22: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P2,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option)))
    <=> ~ ( ( ( Option = none(A) )
            & ~ pp(aa(B,bool,P2,F1)) )
          | ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
            & ~ pp(aa(B,bool,P2,aa(A,B,F22,aa(option(A),A,the2(A),Option)))) ) ) ) ).

% option.split_sel_asm
tff(fact_5388_option_Osplit__sel,axiom,
    ! [B: $tType,A: $tType,P2: fun(B,bool),F1: B,F22: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P2,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option)))
    <=> ( ( ( Option = none(A) )
         => pp(aa(B,bool,P2,F1)) )
        & ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
         => pp(aa(B,bool,P2,aa(A,B,F22,aa(option(A),A,the2(A),Option)))) ) ) ) ).

% option.split_sel
tff(fact_5389_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),set(A),set2(A),nths(A,Xs,I5)) = aa(fun(A,bool),set(A),collect(A),aa(set(nat),fun(A,bool),aTP_Lamp_yl(list(A),fun(set(nat),fun(A,bool)),Xs),I5)) ).

% set_nths
tff(fact_5390_funpow__inj__finite,axiom,
    ! [A: $tType,P: fun(A,A),X: A] :
      ( inj_on(A,A,P,top_top(set(A)))
     => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_ym(fun(A,A),fun(A,fun(A,bool)),P),X))))
       => ~ ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N3),P),X) != X ) ) ) ) ).

% funpow_inj_finite
tff(fact_5391_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,Deg3: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList2,Summary),Deg3)
    <=> ( ( Deg = Deg3 )
        & ! [X4: vEBT_VEBT] :
            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
           => vEBT_VEBT_valid(X4,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
        & vEBT_VEBT_valid(Summary,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
        & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_yb(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg),TreeList2),Summary))),Mima2)) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_5392_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ vEBT_VEBT_valid(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2))
               => ( Xa2 = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList,Summary2)),Xa2))
                 => ( ( Deg2 = Xa2 )
                    & ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => vEBT_VEBT_valid(X3,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
                    & vEBT_VEBT_valid(Summary2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
                    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                    & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_yb(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_5393_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( vEBT_VEBT_valid(X,Xa2)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2))
               => ( Xa2 != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList,Summary2)),Xa2))
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X5: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X5),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                         => vEBT_VEBT_valid(X5,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
                      & vEBT_VEBT_valid(Summary2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_yb(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList),Summary2))),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_5394_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
      ( ( vEBT_VEBT_valid(X,Xa2)
      <=> pp(Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),X),Xa2))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( pp(Y)
                <=> ( Xa2 = one_one(nat) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf(Uu2,Uv2)),Xa2)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList,Summary2) )
               => ( ( pp(Y)
                  <=> ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                         => vEBT_VEBT_valid(X4,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
                      & vEBT_VEBT_valid(Summary2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
                      & pp(aa(option(product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool),aa(bool,fun(fun(product_prod(nat,nat),bool),fun(option(product_prod(nat,nat)),bool)),case_option(bool,product_prod(nat,nat)),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions)))))),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_yb(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList),Summary2))),Mima)) ) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList,Summary2)),Xa2)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_5395_Sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A6: set(set(A))] : aa(set(A),A,complete_Sup_Sup(A),image2(set(A),A,complete_Inf_Inf(A),A6)) = aa(set(A),A,complete_Inf_Inf(A),image2(set(A),A,complete_Sup_Sup(A),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_yn(set(set(A)),fun(set(A),bool),A6)))) ) ).

% Sup_Inf
tff(fact_5396_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(set(A),A,complete_Inf_Inf(A),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_yo(set(set(A)),fun(set(A),bool),A6))))),aa(set(A),A,complete_Inf_Inf(A),image2(set(A),A,complete_Sup_Sup(A),A6)))) ) ).

% Sup_Inf_le
tff(fact_5397_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A6: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(set(A),A,complete_Sup_Sup(A),A6))),aa(set(A),A,complete_Sup_Sup(A),image2(set(A),A,complete_Inf_Inf(A),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_yn(set(set(A)),fun(set(A),bool),A6)))))) ) ).

% Inf_Sup_le
tff(fact_5398_SUP__INF__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G3: fun(B,A),A6: set(set(B))] : aa(set(A),A,complete_Sup_Sup(A),image2(set(B),A,aTP_Lamp_yp(fun(B,A),fun(set(B),A),G3),A6)) = aa(set(A),A,complete_Inf_Inf(A),image2(set(B),A,aTP_Lamp_yq(fun(B,A),fun(set(B),A),G3),aa(fun(set(B),bool),set(set(B)),collect(set(B)),aTP_Lamp_yr(set(set(B)),fun(set(B),bool),A6)))) ) ).

% SUP_INF_set
tff(fact_5399_INF__SUP__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G3: fun(B,A),A6: set(set(B))] : aa(set(A),A,complete_Inf_Inf(A),image2(set(B),A,aTP_Lamp_yq(fun(B,A),fun(set(B),A),G3),A6)) = aa(set(A),A,complete_Sup_Sup(A),image2(set(B),A,aTP_Lamp_yp(fun(B,A),fun(set(B),A),G3),aa(fun(set(B),bool),set(set(B)),collect(set(B)),aTP_Lamp_yr(set(set(B)),fun(set(B),bool),A6)))) ) ).

% INF_SUP_set
tff(fact_5400_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A6: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(set(A),A,complete_Sup_Sup(A),A6))),aa(set(A),A,complete_Sup_Sup(A),image2(set(A),A,complete_Inf_Inf(A),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ys(set(set(A)),fun(set(A),bool),A6)))))) ) ).

% finite_Inf_Sup
tff(fact_5401_lexn__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),N: nat] : lexn(A,R,N) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_yt(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),R),N))) ).

% lexn_conv
tff(fact_5402_lexord__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lexord(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_yu(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).

% lexord_def
tff(fact_5403_lexord__cons__cons,axiom,
    ! [A: $tType,A3: A,X: list(A),B2: A,Y: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y))),lexord(A,R)))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
        | ( ( A3 = B2 )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R))) ) ) ) ).

% lexord_cons_cons
tff(fact_5404_lexn_Osimps_I1_J,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lexn(A,R,zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).

% lexn.simps(1)
tff(fact_5405_lexord__linear,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: list(A),Y: list(A)] :
      ( ! [A5: A,B4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4)),R))
          | ( A5 = B4 )
          | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5)),R)) )
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R)))
        | ( X = Y )
        | pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),X)),lexord(A,R))) ) ) ).

% lexord_linear
tff(fact_5406_lexord__irreflexive,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R))
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lexord(A,R))) ) ).

% lexord_irreflexive
tff(fact_5407_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
      ( ! [X3: A,Y3: A,Z: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R))
           => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R))
             => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z)),R)) ) ) )
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,R)))
       => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lexord(A,R)))
         => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs)),lexord(A,R))) ) ) ) ).

% lexord_partial_trans
tff(fact_5408_lexord__append__leftD,axiom,
    ! [A: $tType,X: list(A),U: list(A),V3: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,X,U)),append(A,X,V3))),lexord(A,R)))
     => ( ! [A5: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5)),R))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V3)),lexord(A,R))) ) ) ).

% lexord_append_leftD
tff(fact_5409_lexord__sufE,axiom,
    ! [A: $tType,Xs: list(A),Zs: list(A),Ys: list(A),Qs: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Zs)),append(A,Ys,Qs))),lexord(A,R)))
     => ( ( Xs != Ys )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
         => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(A),nat,size_size(list(A)),Qs) )
           => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,R))) ) ) ) ) ).

% lexord_sufE
tff(fact_5410_lexord__lex,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lex(A,R)))
    <=> ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R)))
        & ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).

% lexord_lex
tff(fact_5411_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A)),N: nat] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexn(A,R,N)))
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
        & ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ) ).

% lexn_length
tff(fact_5412_lexord__append__left__rightI,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),U: list(A),X: list(A),Y: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,U,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X))),append(A,U,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y)))),lexord(A,R))) ) ).

% lexord_append_left_rightI
tff(fact_5413_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Ys)),append(A,Xs,Zs))),lexord(A,R)))
    <=> ( ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R)) )
        | pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lexord(A,R))) ) ) ).

% lexord_same_pref_iff
tff(fact_5414_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W2: list(A),R: set(product_prod(A,A)),V3: list(A),Z2: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),W2)),lexord(A,R)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W2)),aa(list(A),nat,size_size(list(A)),U)))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,U,V3)),append(A,W2,Z2))),lexord(A,R))) ) ) ).

% lexord_sufI
tff(fact_5415_mlex__eq,axiom,
    ! [A: $tType,F3: fun(A,nat),R2: set(product_prod(A,A))] : mlex_prod(A,F3,R2) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_yv(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),F3),R2))) ).

% mlex_eq
tff(fact_5416_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_lw(A,fun(A,bool))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_5417_apfst__convE,axiom,
    ! [C: $tType,A: $tType,B: $tType,Q2: product_prod(A,B),F3: fun(C,A),P: product_prod(C,B)] :
      ( ( Q2 = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),P) )
     => ~ ! [X3: C,Y3: B] :
            ( ( P = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X3),Y3) )
           => ( Q2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,X3)),Y3) ) ) ) ).

% apfst_convE
tff(fact_5418_mlex__leq,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F3,R2))) ) ) ).

% mlex_leq
tff(fact_5419_mlex__iff,axiom,
    ! [A: $tType,X: A,Y: A,F3: fun(A,nat),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F3,R2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
        | ( ( aa(A,nat,F3,X) = aa(A,nat,F3,Y) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2)) ) ) ) ).

% mlex_iff
tff(fact_5420_mlex__less,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),mlex_prod(A,F3,R2))) ) ).

% mlex_less
tff(fact_5421_in__measure,axiom,
    ! [A: $tType,X: A,Y: A,F3: fun(A,nat)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),measure(A,F3)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y))) ) ).

% in_measure
tff(fact_5422_lex__prod__def,axiom,
    ! [A: $tType,B: $tType,Ra: set(product_prod(A,A)),Rb: set(product_prod(B,B))] : lex_prod(A,B,Ra,Rb) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),bool)),fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),product_case_prod(product_prod(A,B),product_prod(A,B),bool),aa(fun(A,fun(B,fun(product_prod(A,B),bool))),fun(product_prod(A,B),fun(product_prod(A,B),bool)),product_case_prod(A,B,fun(product_prod(A,B),bool)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_yx(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool)))),Ra),Rb)))) ).

% lex_prod_def
tff(fact_5423_image2__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,A6: set(C),F3: fun(C,A),G3: fun(C,B)] : bNF_Greatest_image2(C,A,B,A6,F3,G3) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_yy(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),A6),F3),G3)) ).

% image2_def
tff(fact_5424_in__lex__prod,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B,R: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))),lex_prod(A,B,R,S2)))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A4)),R))
        | ( ( A3 = A4 )
          & pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B3)),S2)) ) ) ) ).

% in_lex_prod
tff(fact_5425_image2__eqI,axiom,
    ! [A: $tType,C: $tType,B: $tType,B2: A,F3: fun(B,A),X: B,C3: C,G3: fun(B,C),A6: set(B)] :
      ( ( B2 = aa(B,A,F3,X) )
     => ( ( C3 = aa(B,C,G3,X) )
       => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A6))
         => pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B2),C3)),bNF_Greatest_image2(B,A,C,A6,F3,G3))) ) ) ) ).

% image2_eqI
tff(fact_5426_same__fst__def,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),R2: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P2,R2) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),bool)),fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),product_case_prod(product_prod(A,B),product_prod(A,B),bool),aa(fun(A,fun(B,fun(product_prod(A,B),bool))),fun(product_prod(A,B),fun(product_prod(A,B),bool)),product_case_prod(A,B,fun(product_prod(A,B),bool)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_za(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),P2),R2)))) ).

% same_fst_def
tff(fact_5427_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: num,N: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M2)),aa(num,A,numeral_numeral(A),N)) = aa(option(num),A,aa(fun(num,A),fun(option(num),A),aa(A,fun(fun(num,A),fun(option(num),A)),case_option(A,num),zero_zero(A)),numeral_numeral(A)),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M2),N)) ) ).

% take_bit_numeral_numeral
tff(fact_5428_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_zb(list(A),fun(list(B),fun(product_prod(A,B),bool)),Xs),Ys)) ).

% set_zip
tff(fact_5429_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% min.bounded_iff
tff(fact_5430_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_5431_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb1
tff(fact_5432_min__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),top_top(A)) = X ) ).

% min_top2
tff(fact_5433_min__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),top_top(A)),X) = X ) ).

% min_top
tff(fact_5434_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),X) = bot_bot(A) ) ).

% min_bot
tff(fact_5435_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),bot_bot(A)) = bot_bot(A) ) ).

% min_bot2
tff(fact_5436_max__min__same_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = X ) ).

% max_min_same(1)
tff(fact_5437_max__min__same_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),X) = X ) ).

% max_min_same(2)
tff(fact_5438_max__min__same_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Y) = Y ) ).

% max_min_same(3)
tff(fact_5439_max__min__same_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Y),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = Y ) ).

% max_min_same(4)
tff(fact_5440_min__Suc__Suc,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M2)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M2),N)) ).

% min_Suc_Suc
tff(fact_5441_min__0L,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% min_0L
tff(fact_5442_min__0R,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_5443_take__bit__num__simps_I1_J,axiom,
    ! [M2: num] : bit_take_bit_num(zero_zero(nat),M2) = none(num) ).

% take_bit_num_simps(1)
tff(fact_5444_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)) = aa(num,A,numeral_numeral(A),V3) ) ) ) ) ).

% min_number_of(1)
tff(fact_5445_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_5446_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_5447_min__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),zero_zero(A)) = zero_zero(A) ) ) ).

% min_0_1(2)
tff(fact_5448_min__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),one_one(A)) = zero_zero(A) ) ) ).

% min_0_1(1)
tff(fact_5449_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = one_one(A) ) ).

% min_0_1(5)
tff(fact_5450_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = one_one(A) ) ).

% min_0_1(6)
tff(fact_5451_Int__atMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),A3)),aa(A,set(A),set_ord_atMost(A),B2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)) ) ).

% Int_atMost
tff(fact_5452_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa(nat,nat,suc,N),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(2)
tff(fact_5453_length__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),take(A,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_take
tff(fact_5454_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M2: num] : bit_take_bit_num(aa(nat,nat,suc,N),bit0(M2)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),none(num)),aTP_Lamp_zc(num,option(num))),bit_take_bit_num(N,M2)) ).

% take_bit_num_simps(3)
tff(fact_5455_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3)) ) ) ) ) ).

% min_number_of(2)
tff(fact_5456_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V3)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V3)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V3)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V3)) = aa(num,A,numeral_numeral(A),V3) ) ) ) ) ).

% min_number_of(3)
tff(fact_5457_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3)) ) ) ) ) ).

% min_number_of(4)
tff(fact_5458_same__fstI,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),X: A,Y7: B,Y: B,R2: fun(A,set(product_prod(B,B)))] :
      ( pp(aa(A,bool,P2,X))
     => ( pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y7),Y)),aa(A,set(product_prod(B,B)),R2,X)))
       => pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y7)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y))),same_fst(A,B,P2,R2))) ) ) ).

% same_fstI
tff(fact_5459_Int__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C3,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastAtMost
tff(fact_5460_Int__atLeastAtMostR1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),B2)),set_or1337092689740270186AtMost(A,C3,D3)) = set_or1337092689740270186AtMost(A,C3,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastAtMostR1
tff(fact_5461_Int__atLeastAtMostL1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(A,set(A),set_ord_atMost(A),D3)) = set_or1337092689740270186AtMost(A,A3,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastAtMostL1
tff(fact_5462_Int__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C3,D3)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastLessThan
tff(fact_5463_Int__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or5935395276787703475ssThan(A,C3,D3)) = set_or5935395276787703475ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_greaterThanLessThan
tff(fact_5464_Int__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C3,D3)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_greaterThanAtMost
tff(fact_5465_min__numeral__Suc,axiom,
    ! [K2: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),pred_numeral(K2)),N)) ).

% min_numeral_Suc
tff(fact_5466_min__Suc__numeral,axiom,
    ! [N: nat,K2: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K2)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),pred_numeral(K2))) ).

% min_Suc_numeral
tff(fact_5467_zip__replicate,axiom,
    ! [A: $tType,B: $tType,I2: nat,X: A,J2: nat,Y: B] : zip(A,B,replicate(A,I2,X),replicate(B,J2,Y)) = replicate(product_prod(A,B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I2),J2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) ).

% zip_replicate
tff(fact_5468_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M2: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit1,M2)) = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_take_bit_num(N,M2))) ).

% take_bit_num_simps(4)
tff(fact_5469_length__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),zip(A,B,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).

% length_zip
tff(fact_5470_greaterThan__Int__greaterThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),A3)),aa(A,set(A),set_ord_lessThan(A),B2)) = aa(A,set(A),set_ord_lessThan(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)) ) ).

% greaterThan_Int_greaterThan
tff(fact_5471_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X5: A,Xa: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X5),Xa) = X5 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X5),Xa) = Xa ) ) ) ) ).

% min_def_raw
tff(fact_5472_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C3),D3))) ) ) ) ).

% min.mono
tff(fact_5473_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).

% min.orderE
tff(fact_5474_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).

% min.orderI
tff(fact_5475_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3)) ) ) ) ).

% min.boundedE
tff(fact_5476_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3))) ) ) ) ).

% min.boundedI
tff(fact_5477_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).

% min.order_iff
tff(fact_5478_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),A3)) ) ).

% min.cobounded1
tff(fact_5479_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2)) ) ).

% min.cobounded2
tff(fact_5480_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) ) ) ).

% min.absorb_iff1
tff(fact_5481_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_5482_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C3)) ) ) ).

% min.coboundedI1
tff(fact_5483_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C3)) ) ) ).

% min.coboundedI2
tff(fact_5484_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).

% min_le_iff_disj
tff(fact_5485_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_5486_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = X ) ) ) ).

% min_absorb1
tff(fact_5487_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = A3 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ) ).

% min_def
tff(fact_5488_min__diff,axiom,
    ! [M2: nat,I2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),I2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M2),N)),I2) ).

% min_diff
tff(fact_5489_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2)) ) ).

% min_diff_distrib_left
tff(fact_5490_nat__mult__min__left,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M2),N)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ).

% nat_mult_min_left
tff(fact_5491_nat__mult__min__right,axiom,
    ! [M2: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),Q2)) ).

% nat_mult_min_right
tff(fact_5492_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_5493_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_5494_min__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_min(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_min(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)) ) ).

% min_add_distrib_left
tff(fact_5495_min__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_min(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2)) ) ).

% min_add_distrib_right
tff(fact_5496_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_min_eq_max
tff(fact_5497_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_max_eq_min
tff(fact_5498_zip__obtain__same__length,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P2: fun(list(product_prod(A,B)),bool)] :
      ( ! [Zs2: list(A),Ws2: list(B),N3: nat] :
          ( ( aa(list(A),nat,size_size(list(A)),Zs2) = aa(list(B),nat,size_size(list(B)),Ws2) )
         => ( ( N3 = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) )
           => ( ( Zs2 = take(A,N3,Xs) )
             => ( ( Ws2 = take(B,N3,Ys) )
               => pp(aa(list(product_prod(A,B)),bool,P2,zip(A,B,Zs2,Ws2))) ) ) ) )
     => pp(aa(list(product_prod(A,B)),bool,P2,zip(A,B,Xs,Ys))) ) ).

% zip_obtain_same_length
tff(fact_5499_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P),Y)) ) ) ) ) ).

% max_mult_distrib_left
tff(fact_5500_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P),Y)) ) ) ) ) ).

% min_mult_distrib_left
tff(fact_5501_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P)) ) ) ) ) ).

% max_mult_distrib_right
tff(fact_5502_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P)) ) ) ) ) ).

% min_mult_distrib_right
tff(fact_5503_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y),P) = aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,X,P)),divide_divide(A,Y,P)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y),P) = aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,X,P)),divide_divide(A,Y,P)) ) ) ) ) ).

% max_divide_distrib_right
tff(fact_5504_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y),P) = aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,X,P)),divide_divide(A,Y,P)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y),P) = aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,X,P)),divide_divide(A,Y,P)) ) ) ) ) ).

% min_divide_distrib_right
tff(fact_5505_min__Suc1,axiom,
    ! [N: nat,M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),M2) = case_nat(nat,zero_zero(nat),aTP_Lamp_zd(nat,fun(nat,nat),N),M2) ).

% min_Suc1
tff(fact_5506_min__Suc2,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M2),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_ze(nat,fun(nat,nat),N),M2) ).

% min_Suc2
tff(fact_5507_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: num] :
          ( ( bit_take_bit_num(M2,N) = none(num) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M2),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5508_map__fst__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : map(product_prod(A,B),A,product_fst(A,B),zip(A,B,Xs,Ys)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)),Xs) ).

% map_fst_zip_take
tff(fact_5509_map__snd__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(B),Ys: list(A)] : map(product_prod(B,A),A,product_snd(B,A),zip(B,A,Xs,Ys)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)),Ys) ).

% map_snd_zip_take
tff(fact_5510_take__bit__num__def,axiom,
    ! [N: nat,M2: num] :
      ( ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M2)) = zero_zero(nat) )
       => ( bit_take_bit_num(N,M2) = none(num) ) )
      & ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M2)) != zero_zero(nat) )
       => ( bit_take_bit_num(N,M2) = aa(num,option(num),some(num),num_of_nat(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M2)))) ) ) ) ).

% take_bit_num_def
tff(fact_5511_lexord__take__index__conv,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),lexord(A,R)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y)))
          & ( take(A,aa(list(A),nat,size_size(list(A)),X),Y) = X ) )
        | ? [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y))))
            & ( take(A,I,X) = take(A,I,Y) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,X),I)),aa(nat,A,nth(A,Y),I))),R)) ) ) ) ).

% lexord_take_index_conv
tff(fact_5512_set__relcomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,Xys: list(product_prod(A,C)),Yzs: list(product_prod(C,B))] : relcomp(A,C,B,aa(list(product_prod(A,C)),set(product_prod(A,C)),set2(product_prod(A,C)),Xys),aa(list(product_prod(C,B)),set(product_prod(C,B)),set2(product_prod(C,B)),Yzs)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),map(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_zg(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs),Xys))) ).

% set_relcomp
tff(fact_5513_ran__map__of__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( distinct(A,Xs)
       => ( ran(A,B,map_of(A,B,zip(A,B,Xs,Ys))) = aa(list(B),set(B),set2(B),Ys) ) ) ) ).

% ran_map_of_zip
tff(fact_5514_has__derivative__power__int_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,N: int,S3: set(A)] :
          ( ( X != zero_zero(A) )
         => has_derivative(A,A,aTP_Lamp_zh(int,fun(A,A),N),aa(int,fun(A,A),aTP_Lamp_zi(A,fun(int,fun(A,A)),X),N),topolo174197925503356063within(A,X,S3)) ) ) ).

% has_derivative_power_int'
tff(fact_5515_power__int__mult__distrib__numeral2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,W2: num,M2: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(num,A,numeral_numeral(A),W2)),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M2)),power_int(A,aa(num,A,numeral_numeral(A),W2),M2)) ) ).

% power_int_mult_distrib_numeral2
tff(fact_5516_power__int__mult__distrib__numeral1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [W2: num,Y: A,M2: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Y),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W2),M2)),power_int(A,Y,M2)) ) ).

% power_int_mult_distrib_numeral1
tff(fact_5517_power__int__0__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M2: int] :
          ( ( M2 != zero_zero(int) )
         => ( power_int(A,zero_zero(A),M2) = zero_zero(A) ) ) ) ).

% power_int_0_left
tff(fact_5518_power__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] :
          ( ( power_int(A,X,N) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( N != zero_zero(int) ) ) ) ) ).

% power_int_eq_0_iff
tff(fact_5519_power__int__mult__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: num,N: num] : power_int(A,power_int(A,X,aa(num,int,numeral_numeral(int),M2)),aa(num,int,numeral_numeral(int),N)) = power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),times_times(num),M2),N))) ) ).

% power_int_mult_numeral
tff(fact_5520_power__int__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M2: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)) = one_one(A) ) ).

% power_int_minus_one_mult_self
tff(fact_5521_power__int__minus__one__mult__self_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M2: int,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M2)),B2)) = B2 ) ).

% power_int_minus_one_mult_self'
tff(fact_5522_power__int__add__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M2))),power_int(A,X,aa(num,int,numeral_numeral(int),N))) = power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N))) ) ).

% power_int_add_numeral
tff(fact_5523_power__int__add__numeral2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M2))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N)))),B2) ) ).

% power_int_add_numeral2
tff(fact_5524_power__int__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N)),power_int(A,B2,N)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_5525_relcompEpair,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: A,C3: B,R: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),C3)),relcomp(A,C,B,R,S2)))
     => ~ ! [B4: C] :
            ( pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B4)),R))
           => ~ pp(aa(set(product_prod(C,B)),bool,aa(product_prod(C,B),fun(set(product_prod(C,B)),bool),member(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B4),C3)),S2)) ) ) ).

% relcompEpair
tff(fact_5526_relcompE,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),Xz),relcomp(A,C,B,R,S2)))
     => ~ ! [X3: A,Y3: C,Z: B] :
            ( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z) )
           => ( pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X3),Y3)),R))
             => ~ pp(aa(set(product_prod(C,B)),bool,aa(product_prod(C,B),fun(set(product_prod(C,B)),bool),member(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y3),Z)),S2)) ) ) ) ).

% relcompE
tff(fact_5527_relcomp_OrelcompI,axiom,
    ! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(A,B)),C3: C,S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
     => ( pp(aa(set(product_prod(B,C)),bool,aa(product_prod(B,C),fun(set(product_prod(B,C)),bool),member(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C3)),S2))
       => pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C3)),relcomp(A,B,C,R,S2))) ) ) ).

% relcomp.relcompI
tff(fact_5528_relcomp_Osimps,axiom,
    ! [A: $tType,C: $tType,B: $tType,A12: A,A23: C,R: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A12),A23)),relcomp(A,B,C,R,S2)))
    <=> ? [A7: A,B5: B,C4: C] :
          ( ( A12 = A7 )
          & ( A23 = C4 )
          & pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5)),R))
          & pp(aa(set(product_prod(B,C)),bool,aa(product_prod(B,C),fun(set(product_prod(B,C)),bool),member(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B5),C4)),S2)) ) ) ).

% relcomp.simps
tff(fact_5529_relcomp_Ocases,axiom,
    ! [A: $tType,C: $tType,B: $tType,A12: A,A23: C,R: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A12),A23)),relcomp(A,B,C,R,S2)))
     => ~ ! [B4: B] :
            ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A12),B4)),R))
           => ~ pp(aa(set(product_prod(B,C)),bool,aa(product_prod(B,C),fun(set(product_prod(B,C)),bool),member(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),A23)),S2)) ) ) ).

% relcomp.cases
tff(fact_5530_power__int__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,M2: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y),M2) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M2)),power_int(A,Y,M2)) ) ).

% power_int_mult_distrib
tff(fact_5531_power__int__commutes,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,N)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,N)) ) ).

% power_int_commutes
tff(fact_5532_power__int__mult,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: int,N: int] : power_int(A,X,aa(int,int,aa(int,fun(int,int),times_times(int),M2),N)) = power_int(A,power_int(A,X,M2),N) ) ).

% power_int_mult
tff(fact_5533_ranI,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),M2,A3) = aa(A,option(A),some(A),B2) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),ran(B,A,M2))) ) ).

% ranI
tff(fact_5534_zero__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),power_int(A,X,N))) ) ) ).

% zero_less_power_int
tff(fact_5535_power__int__not__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( N = zero_zero(int) ) )
         => ( power_int(A,X,N) != zero_zero(A) ) ) ) ).

% power_int_not_zero
tff(fact_5536_zero__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),power_int(A,X,N))) ) ) ).

% zero_le_power_int
tff(fact_5537_continuous__on__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S2: set(A),F3: fun(A,B),N: int] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => ( aa(A,B,F3,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S2,aa(int,fun(A,B),aTP_Lamp_zj(fun(A,B),fun(int,fun(A,B)),F3),N)) ) ) ) ).

% continuous_on_power_int
tff(fact_5538_power__int__0__left__If,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M2: int] :
          ( ( ( M2 = zero_zero(int) )
           => ( power_int(A,zero_zero(A),M2) = one_one(A) ) )
          & ( ( M2 != zero_zero(int) )
           => ( power_int(A,zero_zero(A),M2) = zero_zero(A) ) ) ) ) ).

% power_int_0_left_If
tff(fact_5539_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N6: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N)),power_int(A,A3,N6))) ) ) ) ).

% power_int_increasing
tff(fact_5540_power__int__diff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,M2: int,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( M2 != N ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),M2),N)) = divide_divide(A,power_int(A,X,M2),power_int(A,X,N)) ) ) ) ).

% power_int_diff
tff(fact_5541_tendsto__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(B,A),A3: A,F4: filter(B),N: int] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,A3),F4)
         => ( ( A3 != zero_zero(A) )
           => filterlim(B,A,aa(int,fun(B,A),aTP_Lamp_zk(fun(B,A),fun(int,fun(B,A)),F3),N),topolo7230453075368039082e_nhds(A,power_int(A,A3,N)),F4) ) ) ) ).

% tendsto_power_int
tff(fact_5542_continuous__at__within__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [A3: A,S2: set(A),F3: fun(A,B),N: int] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S2),F3)
         => ( ( aa(A,B,F3,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S2),aa(int,fun(A,B),aTP_Lamp_zl(fun(A,B),fun(int,fun(A,B)),F3),N)) ) ) ) ).

% continuous_at_within_power_int
tff(fact_5543_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: set(product_prod(A,C)),S2: set(product_prod(C,B))] : relcomp(A,C,B,R,S2) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_zm(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),R),S2))) ).

% relcomp_unfold
tff(fact_5544_differentiable__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),X: A,S2: set(A),N: int] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,X,S2))
         => ( ( aa(A,B,F3,X) != zero_zero(B) )
           => differentiable(A,B,aa(int,fun(A,B),aTP_Lamp_zn(fun(A,B),fun(int,fun(A,B)),F3),N),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% differentiable_power_int
tff(fact_5545_ran__def,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] : ran(A,B,M2) = aa(fun(B,bool),set(B),collect(B),aTP_Lamp_zo(fun(A,option(B)),fun(B,bool),M2)) ).

% ran_def
tff(fact_5546_continuous__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F3: fun(A,B),N: int] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( ( aa(A,B,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_uw(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aa(int,fun(A,B),aTP_Lamp_zl(fun(A,B),fun(int,fun(A,B)),F3),N)) ) ) ) ).

% continuous_power_int
tff(fact_5547_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N6: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N6)),power_int(A,A3,N))) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_5548_power__int__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),power_int(A,Y,N))) ) ) ) ) ).

% power_int_mono
tff(fact_5549_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),zero_zero(int)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,B2,N)),power_int(A,A3,N))) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_5550_one__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),power_int(A,X,N))) ) ) ) ).

% one_le_power_int
tff(fact_5551_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: int,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),N) != zero_zero(int) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M2)),power_int(A,X,N)) ) ) ) ).

% power_int_add
tff(fact_5552_power__int__minus__left__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( division_ring(A)
        & one(B)
        & uminus(B) )
     => ! [X: C,A3: A,N: int] :
          ( nO_MATCH(B,C,aa(B,B,uminus_uminus(B),one_one(B)),X)
         => ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),N)),power_int(A,A3,N)) ) ) ) ).

% power_int_minus_left_distrib
tff(fact_5553_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),zero_zero(int)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,B2,N)),power_int(A,A3,N))) ) ) ) ) ).

% power_int_antimono
tff(fact_5554_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N)),power_int(A,B2,N))) ) ) ) ) ).

% power_int_strict_mono
tff(fact_5555_power__int__le__one,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),one_one(A))) ) ) ) ) ).

% power_int_le_one
tff(fact_5556_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N6: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
             => ( ( ( A3 != zero_zero(A) )
                  | ( N6 != zero_zero(int) )
                  | ( N = zero_zero(int) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N6)),power_int(A,A3,N))) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_5557_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M2: int,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,M2)),power_int(A,X,N)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M2),N)) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_5558_power__int__minus__mult,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( N != zero_zero(int) ) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),N),one_one(int)))),X) = power_int(A,X,N) ) ) ) ).

% power_int_minus_mult
tff(fact_5559_power__int__add__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: int] :
          ( ( ( X != zero_zero(A) )
            | ( M2 != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M2)),X) ) ) ) ).

% power_int_add_1
tff(fact_5560_power__int__add__1_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M2: int] :
          ( ( ( X != zero_zero(A) )
            | ( M2 != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M2),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,M2)) ) ) ) ).

% power_int_add_1'
tff(fact_5561_DERIV__power__int,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D3: A,X: A,S2: set(A),N: int] :
          ( has_field_derivative(A,F3,D3,topolo174197925503356063within(A,X,S2))
         => ( ( aa(A,A,F3,X) != zero_zero(A) )
           => has_field_derivative(A,aa(int,fun(A,A),aTP_Lamp_zp(fun(A,A),fun(int,fun(A,A)),F3),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),N)),power_int(A,aa(A,A,F3,X),aa(int,int,aa(int,fun(int,int),minus_minus(int),N),one_one(int))))),D3),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_power_int
tff(fact_5562_has__derivative__power__int,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [F3: fun(C,A),X: C,F6: fun(C,A),S3: set(C),N: int] :
          ( ( aa(C,A,F3,X) != zero_zero(A) )
         => ( has_derivative(C,A,F3,F6,topolo174197925503356063within(C,X,S3))
           => has_derivative(C,A,aa(int,fun(C,A),aTP_Lamp_zq(fun(C,A),fun(int,fun(C,A)),F3),N),aa(int,fun(C,A),aa(fun(C,A),fun(int,fun(C,A)),aa(C,fun(fun(C,A),fun(int,fun(C,A))),aTP_Lamp_zr(fun(C,A),fun(C,fun(fun(C,A),fun(int,fun(C,A)))),F3),X),F6),N),topolo174197925503356063within(C,X,S3)) ) ) ) ).

% has_derivative_power_int
tff(fact_5563_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = case_list(list(product_prod(A,B)),B,nil(product_prod(A,B)),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_zs(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X),Xs),Ys) ).

% zip_Cons1
tff(fact_5564_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = case_list(list(product_prod(A,B)),A,nil(product_prod(A,B)),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_zt(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys),Xs) ).

% zip_Cons
tff(fact_5565_pred__nat__def,axiom,
    pred_nat = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_zu(nat,fun(nat,bool)))) ).

% pred_nat_def
tff(fact_5566_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A3)) )
           => ( linorder_insort_key(A,A,aTP_Lamp_ni(A,A),A3,Xs) = append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_5567_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B6: set(A),X: B,A6: set(product_prod(B,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),B6))
     => ( 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))),image2(A,set(product_prod(B,A)),aTP_Lamp_zv(B,fun(A,set(product_prod(B,A))),X),B6))),A6) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_zw(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),X),A6,B6) ) ) ).

% fold_union_pair
tff(fact_5568_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A6) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_zx(A,fun(option(A),option(A))),none(A),A6)) ) ).

% Min.eq_fold'
tff(fact_5569_length__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),linorder_insort_key(B,A,F3,X,Xs)) = aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),Xs)) ) ).

% length_insort
tff(fact_5570_insort__key_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Y: B,Ys: list(B)] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X)),aa(B,A,F3,Y)))
           => ( linorder_insort_key(B,A,F3,X,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X)),aa(B,A,F3,Y)))
           => ( linorder_insort_key(B,A,F3,X,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),linorder_insort_key(B,A,F3,X,Ys)) ) ) ) ) ).

% insort_key.simps(2)
tff(fact_5571_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),linorder_insort_key(A,A,aTP_Lamp_ni(A,A),X,Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_5572_insort__is__Cons,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Xs: list(B),F3: fun(B,A),A3: B] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(list(B),set(B),set2(B),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,A3)),aa(B,A,F3,X3))) )
         => ( linorder_insort_key(B,A,F3,A3,Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_5573_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F3,linorder_insort_key(B,A,F3,X,Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),map(B,A,F3,Xs)) ) ) ).

% sorted_insort_key
tff(fact_5574_relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R2))
     => ( pp(aa(set(product_prod(B,C)),bool,finite_finite2(product_prod(B,C)),S3))
       => ( relcomp(A,B,C,R2,S3) = finite_fold(product_prod(A,B),set(product_prod(A,C)),aa(fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(A,B),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(A,B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_zz(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S3)),bot_bot(set(product_prod(A,C))),R2) ) ) ) ).

% relcomp_fold
tff(fact_5575_insert__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S3: set(product_prod(A,B)),X: product_prod(C,A),R2: set(product_prod(C,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S3))
     => ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),insert(product_prod(C,A),X),R2),S3) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_aaa(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),relcomp(C,A,B,R2,S3),S3) ) ) ).

% insert_relcomp_fold
tff(fact_5576_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P2: fun(B,bool),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F3,Xs))
         => ( pp(aa(B,bool,P2,X))
           => ( filter2(B,P2,linorder_insort_key(B,A,F3,X,Xs)) = linorder_insort_key(B,A,F3,X,filter2(B,P2,Xs)) ) ) ) ) ).

% filter_insort
tff(fact_5577_insert__relcomp__union__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S3: set(product_prod(A,B)),X: product_prod(C,A),X6: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S3))
     => ( aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),insert(product_prod(C,A),X),bot_bot(set(product_prod(C,A)))),S3)),X6) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_aaa(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),X6,S3) ) ) ).

% insert_relcomp_union_fold
tff(fact_5578_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A6: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A6) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_aab(A,fun(option(A),option(A))),none(A),A6)) ) ).

% Max.eq_fold'
tff(fact_5579_insort__key__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [A3: B,Xs: list(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),aa(list(B),set(B),set2(B),Xs)))
         => ( sorted_wrt(A,ord_less_eq(A),map(B,A,F3,Xs))
           => ( ( hd(B,filter2(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_aac(B,fun(fun(B,A),fun(B,bool)),A3),F3),Xs)) = A3 )
             => ( linorder_insort_key(B,A,F3,A3,remove1(B,A3,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_5580_Id__on__fold,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( id_on(A,A6) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_aad(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A6) ) ) ).

% Id_on_fold
tff(fact_5581_Id__on__def,axiom,
    ! [A: $tType,A6: set(A)] : id_on(A,A6) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image2(A,set(product_prod(A,A)),aTP_Lamp_aae(A,set(product_prod(A,A))),A6)) ).

% Id_on_def
tff(fact_5582_Id__onI,axiom,
    ! [A: $tType,A3: A,A6: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),id_on(A,A6))) ) ).

% Id_onI
tff(fact_5583_Id__on__iff,axiom,
    ! [A: $tType,X: A,Y: A,A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),id_on(A,A6)))
    <=> ( ( X = Y )
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6)) ) ) ).

% Id_on_iff
tff(fact_5584_Id__on__eqI,axiom,
    ! [A: $tType,A3: A,B2: A,A6: set(A)] :
      ( ( A3 = B2 )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id_on(A,A6))) ) ) ).

% Id_on_eqI
tff(fact_5585_Id__onE,axiom,
    ! [A: $tType,C3: product_prod(A,A),A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),C3),id_on(A,A6)))
     => ~ ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
           => ( C3 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ) ) ).

% Id_onE
tff(fact_5586_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,A3,Xs)) ) ) ).

% sorted_remove1
tff(fact_5587_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F3,Xs))
         => sorted_wrt(A,ord_less_eq(A),map(B,A,F3,remove1(B,X,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_5588_length__remove1,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% length_remove1
tff(fact_5589_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),aa(list(A),set(A),set2(A),Xs)))
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ( linorder_insort_key(A,A,aTP_Lamp_ni(A,A),A3,remove1(A,A3,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_5590_Id__on__set,axiom,
    ! [A: $tType,Xs: list(A)] : id_on(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_mq(A,product_prod(A,A)),Xs)) ).

% Id_on_set
tff(fact_5591_comp__fun__commute__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S3))
     => finite6289374366891150609ommute(product_prod(C,A),set(product_prod(C,B)),aa(fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(C,A),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(C,A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_aag(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S3))) ) ).

% comp_fun_commute_relcomp_fold
tff(fact_5592_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A)] : lattic5882676163264333800up_fin(A,A6) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_aah(A,fun(option(A),option(A))),none(A),A6)) ) ).

% Sup_fin.eq_fold'
tff(fact_5593_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A)] : lattic7752659483105999362nf_fin(A,A6) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_aai(A,fun(option(A),option(A))),none(A),A6)) ) ).

% Inf_fin.eq_fold'
tff(fact_5594_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic7752659483105999362nf_fin(A,A6)),lattic5882676163264333800up_fin(A,A6))) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_5595_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),lattic5882676163264333800up_fin(A,A6))) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_5596_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic7752659483105999362nf_fin(A,A6)),A3)) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_5597_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic5882676163264333800up_fin(A,A6)),X))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_5598_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),lattic7752659483105999362nf_fin(A,A6)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_5599_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic5882676163264333800up_fin(A,A6)),X)) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_5600_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic5882676163264333800up_fin(A,A6)),X))
             => ! [A8: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A8),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A8),X)) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_5601_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A5)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),lattic7752659483105999362nf_fin(A,A6))) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_5602_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),lattic7752659483105999362nf_fin(A,A6)))
             => ! [A8: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A8),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A8)) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_5603_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A6: set(A),B6: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic5882676163264333800up_fin(A,A6)),lattic5882676163264333800up_fin(A,B6))) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_5604_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A6: set(A),B6: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
         => ( ( A6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite2(A),B6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),lattic7752659483105999362nf_fin(A,B6)),lattic7752659483105999362nf_fin(A,A6))) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_5605_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),B6))
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_aaj(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B6)) ) ).

% comp_fun_commute_product_fold
tff(fact_5606_relpow__finite__bounded1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),K2: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K2),R2)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aak(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aal(set(product_prod(A,A)),fun(nat,bool),R2)))))) ) ) ).

% relpow_finite_bounded1
tff(fact_5607_Rats__eq__int__div__nat,axiom,
    field_char_0_Rats(real) = aa(fun(real,bool),set(real),collect(real),aTP_Lamp_aam(real,bool)) ).

% Rats_eq_int_div_nat
tff(fact_5608_lim__at__infinity__0,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),L: A] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,L),at_infinity(A))
        <=> filterlim(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,F3),inverse_inverse(A)),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% lim_at_infinity_0
tff(fact_5609_fst__comp__apfst,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),C),comp(product_prod(C,B),C,product_prod(A,B),product_fst(C,B)),product_apfst(A,C,B,F3)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F3),product_fst(A,B)) ).

% fst_comp_apfst
tff(fact_5610_snd__comp__apfst,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),B),comp(product_prod(C,B),B,product_prod(A,B),product_snd(C,B)),product_apfst(A,C,B,F3)) = product_snd(A,B) ).

% snd_comp_apfst
tff(fact_5611_fst__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),comp(product_prod(A,C),A,product_prod(A,B),product_fst(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3)) = product_fst(A,B) ).

% fst_comp_apsnd
tff(fact_5612_snd__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),C),comp(product_prod(A,C),C,product_prod(A,B),product_snd(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),comp(B,C,product_prod(A,B),F3),product_snd(A,B)) ).

% snd_comp_apsnd
tff(fact_5613_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,bool),F3: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),filter2(A,P2,map(B,A,F3,Xs))) = aa(list(B),nat,size_size(list(B)),filter2(B,aa(fun(B,A),fun(B,bool),comp(A,bool,B,P2),F3),Xs)) ).

% length_filter_map
tff(fact_5614_finite__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).

% finite_relpow
tff(fact_5615_funpow__Suc__right,axiom,
    ! [A: $tType,N: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)),F3) ).

% funpow_Suc_right
tff(fact_5616_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,F3),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)) ).

% funpow.simps(2)
tff(fact_5617_funpow__add,axiom,
    ! [A: $tType,M2: nat,N: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M2),F3)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)) ).

% funpow_add
tff(fact_5618_apsnd__compose,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F3: fun(C,B),G3: fun(D,C),X: product_prod(A,D)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(product_prod(A,D),product_prod(A,C),aa(fun(D,C),fun(product_prod(A,D),product_prod(A,C)),product_apsnd(D,C,A),G3),X)) = aa(product_prod(A,D),product_prod(A,B),aa(fun(D,B),fun(product_prod(A,D),product_prod(A,B)),product_apsnd(D,B,A),aa(fun(D,C),fun(D,B),comp(C,B,D,F3),G3)),X) ).

% apsnd_compose
tff(fact_5619_Rats__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field_char_0_Rats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field_char_0_Rats(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),field_char_0_Rats(A))) ) ) ) ).

% Rats_mult
tff(fact_5620_card_Ocomp__fun__commute__on,axiom,
    aa(fun(nat,nat),fun(nat,nat),comp(nat,nat,nat,suc),suc) = aa(fun(nat,nat),fun(nat,nat),comp(nat,nat,nat,suc),suc) ).

% card.comp_fun_commute_on
tff(fact_5621_comp__funpow,axiom,
    ! [B: $tType,A: $tType,N: nat,F3: fun(A,A)] : aa(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A)),aa(nat,fun(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A))),compow(fun(fun(B,A),fun(B,A))),N),comp(A,A,B,F3)) = comp(A,A,B,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)) ).

% comp_funpow
tff(fact_5622_Rats__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),field_char_0_Rats(A))) ) ).

% Rats_0
tff(fact_5623_apfst__compose,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(C,A),G3: fun(D,C),X: product_prod(D,B)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(product_prod(D,B),product_prod(C,B),product_apfst(D,C,B,G3),X)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,aa(fun(D,C),fun(D,A),comp(C,A,D,F3),G3)),X) ).

% apfst_compose
tff(fact_5624_snd__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_snd(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_aan(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_5625_fst__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_fst(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_mq(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_5626_snd__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_snd(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_mq(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_5627_fst__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_fst(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_aan(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_5628_relpow__Suc__D2_H,axiom,
    ! [A: $tType,N: nat,R2: set(product_prod(A,A)),X5: A,Y4: A,Z4: A] :
      ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
        & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)),R2)) )
     => ? [W: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),W)),R2))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W),Z4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).

% relpow_Suc_D2'
tff(fact_5629_relpow__Suc__I2,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z2: A,N: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R2))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2))) ) ) ).

% relpow_Suc_I2
tff(fact_5630_relpow__Suc__E2,axiom,
    ! [A: $tType,X: A,Z2: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2)))
     => ~ ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R2))
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).

% relpow_Suc_E2
tff(fact_5631_relpow__Suc__D2,axiom,
    ! [A: $tType,X: A,Z2: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2)))
     => ? [Y3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R2))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).

% relpow_Suc_D2
tff(fact_5632_relpow__Suc__I,axiom,
    ! [A: $tType,X: A,Y: A,N: nat,R2: set(product_prod(A,A)),Z2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R2))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2))) ) ) ).

% relpow_Suc_I
tff(fact_5633_relpow__Suc__E,axiom,
    ! [A: $tType,X: A,Z2: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2)))
     => ~ ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),R2)) ) ) ).

% relpow_Suc_E
tff(fact_5634_relpow__0__E,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2)))
     => ( X = Y ) ) ).

% relpow_0_E
tff(fact_5635_relpow__0__I,axiom,
    ! [A: $tType,X: A,R2: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2))) ).

% relpow_0_I
tff(fact_5636_relpow_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2) = relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2),R2) ).

% relpow.simps(2)
tff(fact_5637_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G3: fun(C,B),A6: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X3: B,Y3: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X3)),aa(B,A,H,Y3))
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,B),fun(C,A),comp(B,A,C,H),G3)),A6) = aa(B,A,H,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G3),A6)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_5638_relpowp__relpow__eq,axiom,
    ! [A: $tType,N: nat,R2: set(product_prod(A,A)),X5: A,Xa: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R2)),X5),Xa))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ).

% relpowp_relpow_eq
tff(fact_5639_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_5640_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_5641_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M2: nat,K2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_5642_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M2: nat,K2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_5643_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_5644_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_5645_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M2: nat,K2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_5646_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M2: nat,K2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_5647_relpow__E,axiom,
    ! [A: $tType,X: A,Z2: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z2 ) )
       => ~ ! [Y3: A,M: nat] :
              ( ( N = aa(nat,nat,suc,M) )
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M),R2)))
               => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),R2)) ) ) ) ) ).

% relpow_E
tff(fact_5648_relpow__E2,axiom,
    ! [A: $tType,X: A,Z2: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z2 ) )
       => ~ ! [Y3: A,M: nat] :
              ( ( N = aa(nat,nat,suc,M) )
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R2))
               => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M),R2))) ) ) ) ) ).

% relpow_E2
tff(fact_5649_case__prod__comp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F3: fun(D,fun(C,A)),G3: fun(B,D),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),aa(fun(B,D),fun(B,fun(C,A)),comp(D,fun(C,A),B,F3),G3)),X) = aa(C,A,aa(D,fun(C,A),F3,aa(B,D,G3,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_5650_relpow__empty,axiom,
    ! [A: $tType,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).

% relpow_empty
tff(fact_5651_sum_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A6: set(B),H: fun(B,C),G3: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( ! [X3: B,Y3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
               => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y3),A6))
                 => ( ( X3 != Y3 )
                   => ( ( aa(B,C,H,X3) = aa(B,C,H,Y3) )
                     => ( aa(C,A,G3,aa(B,C,H,X3)) = zero_zero(A) ) ) ) ) )
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G3),image2(B,C,H,A6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,C),fun(B,A),comp(C,A,B,G3),H)),A6) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_5652_DERIV__image__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Da: A,G3: fun(A,A),X: A,S2: set(A),Db: A] :
          ( has_field_derivative(A,F3,Da,topolo174197925503356063within(A,aa(A,A,G3,X),image2(A,A,G3,S2)))
         => ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,X,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F3),G3),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_image_chain
tff(fact_5653_DERIV__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Da: A,G3: fun(A,A),X: A,Db: A,S2: set(A)] :
          ( has_field_derivative(A,F3,Da,topolo174197925503356063within(A,aa(A,A,G3,X),top_top(set(A))))
         => ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,X,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F3),G3),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,S2)) ) ) ) ).

% DERIV_chain
tff(fact_5654_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),A6: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A6) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,plus_plus(A)),G3),zero_zero(A),A6) ) ).

% sum.eq_fold
tff(fact_5655_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),A6: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A6) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,times_times(A)),G3),one_one(A),A6) ) ).

% prod.eq_fold
tff(fact_5656_snd__fst__flip,axiom,
    ! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,aa(fun(product_prod(B,A),product_prod(A,B)),fun(product_prod(B,A),A),comp(product_prod(A,B),A,product_prod(B,A),product_fst(A,B)),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_mv(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_5657_fst__snd__flip,axiom,
    ! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),A),comp(product_prod(B,A),A,product_prod(A,B),product_snd(B,A)),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_xi(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_5658_relpow__fun__conv,axiom,
    ! [A: $tType,A3: A,B2: A,N: nat,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
    <=> ? [F5: fun(nat,A)] :
          ( ( aa(nat,A,F5,zero_zero(nat)) = A3 )
          & ( aa(nat,A,F5,N) = B2 )
          & ! [I: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
             => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F5,I)),aa(nat,A,F5,aa(nat,nat,suc,I)))),R2)) ) ) ) ).

% relpow_fun_conv
tff(fact_5659_sum__image__le,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(C),G3: fun(A,B),F3: fun(C,A)] :
          ( pp(aa(set(C),bool,finite_finite2(C),I5))
         => ( ! [I3: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),I3),I5))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,G3,aa(C,A,F3,I3)))) )
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),image2(C,A,F3,I5))),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(C,A),fun(C,B),comp(A,B,C,G3),F3)),I5))) ) ) ) ).

% sum_image_le
tff(fact_5660_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_5661_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_5662_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_5663_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_5664_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M2))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_5665_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M2))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_5666_sum_OUnion__comp,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B6: set(set(B)),G3: fun(B,A)] :
          ( ! [X3: set(B)] :
              ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X3),B6))
             => pp(aa(set(B),bool,finite_finite2(B),X3)) )
         => ( ! [A13: set(B)] :
                ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A13),B6))
               => ! [A24: set(B)] :
                    ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A24),B6))
                   => ( ( A13 != A24 )
                     => ! [X3: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A13))
                         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A24))
                           => ( aa(B,A,G3,X3) = zero_zero(A) ) ) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B6)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7311177749621191930dd_sum(set(B),A)),groups7311177749621191930dd_sum(B,A)),G3),B6) ) ) ) ) ).

% sum.Union_comp
tff(fact_5667_sum_OatLeastAtMost__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_add(A)
        & ord(B) )
     => ! [H: fun(nat,B),M2: nat,N: nat,G3: fun(B,A)] :
          ( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M2,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,M2),aa(nat,B,H,N)))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),set_or1337092689740270186AtMost(B,aa(nat,B,H,M2),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,B),fun(nat,A),comp(B,A,nat,G3),H)),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ) ).

% sum.atLeastAtMost_reindex
tff(fact_5668_sum_OatLeastLessThan__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_add(A)
        & ord(B) )
     => ! [H: fun(nat,B),M2: nat,N: nat,G3: fun(B,A)] :
          ( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M2,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,M2),aa(nat,B,H,N)))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),set_or7035219750837199246ssThan(B,aa(nat,B,H,M2),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,B),fun(nat,A),comp(B,A,nat,G3),H)),set_or7035219750837199246ssThan(nat,M2,N)) ) ) ) ).

% sum.atLeastLessThan_reindex
tff(fact_5669_prod_OatLeastAtMost__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(A)
        & ord(B) )
     => ! [H: fun(nat,B),M2: nat,N: nat,G3: fun(B,A)] :
          ( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M2,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,M2),aa(nat,B,H,N)))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),set_or1337092689740270186AtMost(B,aa(nat,B,H,M2),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,B),fun(nat,A),comp(B,A,nat,G3),H)),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ) ).

% prod.atLeastAtMost_reindex
tff(fact_5670_prod_OatLeastLessThan__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(A)
        & ord(B) )
     => ! [H: fun(nat,B),M2: nat,N: nat,G3: fun(B,A)] :
          ( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M2,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,M2),aa(nat,B,H,N)))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),set_or7035219750837199246ssThan(B,aa(nat,B,H,M2),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,B),fun(nat,A),comp(B,A,nat,G3),H)),set_or7035219750837199246ssThan(nat,M2,N)) ) ) ) ).

% prod.atLeastLessThan_reindex
tff(fact_5671_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_mw(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_5672_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_mw(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_5673_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_mw(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_5674_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M2: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_mw(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_5675_relpow__finite__bounded,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),K2: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K2),R2)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aak(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aap(set(product_prod(A,A)),fun(nat,bool),R2)))))) ) ).

% relpow_finite_bounded
tff(fact_5676_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(int,A),M2: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G3),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M2),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G3),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_5677_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(int,A),M2: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G3),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M2),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G3),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M2,N)) ) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_5678_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(int,A),M2: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G3),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M2),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G3),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% sum.atLeast_int_lessThan_int_shift
tff(fact_5679_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M2))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_5680_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M2: nat,N: nat,G3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M2))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_5681_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(int,A),M2: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G3),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M2),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G3),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M2,N)) ) ).

% prod.atLeast_int_lessThan_int_shift
tff(fact_5682_ntrancl__def,axiom,
    ! [A: $tType,N: nat,R2: set(product_prod(A,A))] : transitive_ntrancl(A,N,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aak(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aaq(nat,fun(nat,bool),N)))) ).

% ntrancl_def
tff(fact_5683_divmod__integer__eq__cases,axiom,
    ! [K2: code_integer,L: code_integer] :
      ( ( ( K2 = zero_zero(code_integer) )
       => ( code_divmod_integer(K2,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
      & ( ( K2 != zero_zero(code_integer) )
       => ( ( ( L = zero_zero(code_integer) )
           => ( code_divmod_integer(K2,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K2) ) )
          & ( ( L != zero_zero(code_integer) )
           => ( code_divmod_integer(K2,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,aa(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer)),times_times(code_integer))),sgn_sgn(code_integer)),L),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),aa(code_integer,code_integer,sgn_sgn(code_integer),K2)),aa(code_integer,code_integer,sgn_sgn(code_integer),L)),code_divmod_abs(K2,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_aar(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K2,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_5684_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( transitive_trancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aak(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aal(set(product_prod(A,A)),fun(nat,bool),R2)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_5685_ntrancl__Zero,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : transitive_ntrancl(A,zero_zero(nat),R2) = R2 ).

% ntrancl_Zero
tff(fact_5686_converse__trancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P2: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)))
     => ( ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),B2)),R))
           => pp(aa(A,bool,P2,Y3)) )
       => ( ! [Y3: A,Z: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z),B2)),transitive_trancl(A,R)))
               => ( pp(aa(A,bool,P2,Z))
                 => pp(aa(A,bool,P2,Y3)) ) ) )
         => pp(aa(A,bool,P2,A3)) ) ) ) ).

% converse_trancl_induct
tff(fact_5687_trancl__trans__induct,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A)),P2: fun(A,fun(A,bool))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R)))
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R))
           => pp(aa(A,bool,aa(A,fun(A,bool),P2,X3),Y3)) )
       => ( ! [X3: A,Y3: A,Z: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),transitive_trancl(A,R)))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),P2,X3),Y3))
               => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),transitive_trancl(A,R)))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),P2,Y3),Z))
                   => pp(aa(A,bool,aa(A,fun(A,bool),P2,X3),Z)) ) ) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),P2,X),Y)) ) ) ) ).

% trancl_trans_induct
tff(fact_5688_trancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),transitive_trancl(A,R)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R))) ) ) ).

% trancl_into_trancl2
tff(fact_5689_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),R))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R))) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_5690_irrefl__trancl__rD,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Y: A] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),transitive_trancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_5691_converse__tranclE,axiom,
    ! [A: $tType,X: A,Z2: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_trancl(A,R)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),R))
       => ~ ! [Y3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R))
             => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),transitive_trancl(A,R))) ) ) ) ).

% converse_tranclE
tff(fact_5692_r__r__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R2))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),R2))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R2))) ) ) ).

% r_r_into_trancl
tff(fact_5693_trancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P2: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)))
     => ( ! [Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y3)),R))
           => pp(aa(A,bool,P2,Y3)) )
       => ( ! [Y3: A,Z: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y3)),transitive_trancl(A,R)))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R))
               => ( pp(aa(A,bool,P2,Y3))
                 => pp(aa(A,bool,P2,Z)) ) ) )
         => pp(aa(A,bool,P2,B2)) ) ) ) ).

% trancl_induct
tff(fact_5694_trancl__trans,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A)),Z2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_trancl(A,R)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_trancl(A,R))) ) ) ).

% trancl_trans
tff(fact_5695_tranclE,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => ~ ! [C2: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R)))
             => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C2),B2)),R)) ) ) ) ).

% tranclE
tff(fact_5696_trancl_Or__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R))) ) ).

% trancl.r_into_trancl
tff(fact_5697_trancl_Osimps,axiom,
    ! [A: $tType,A12: A,A23: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),A23)),transitive_trancl(A,R)))
    <=> ( ? [A7: A,B5: A] :
            ( ( A12 = A7 )
            & ( A23 = B5 )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B5)),R)) )
        | ? [A7: A,B5: A,C4: A] :
            ( ( A12 = A7 )
            & ( A23 = C4 )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B5)),transitive_trancl(A,R)))
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),C4)),R)) ) ) ) ).

% trancl.simps
tff(fact_5698_trancl_Ocases,axiom,
    ! [A: $tType,A12: A,A23: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),A23)),transitive_trancl(A,R)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),A23)),R))
       => ~ ! [B4: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),B4)),transitive_trancl(A,R)))
             => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A23)),R)) ) ) ) ).

% trancl.cases
tff(fact_5699_trancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),P2: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_trancl(product_prod(A,B),R)))
     => ( ! [A5: A,B4: B] :
            ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),R))
           => pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) )
       => ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),transitive_trancl(product_prod(A,B),R)))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By)) ) ) ) ).

% trancl_induct2
tff(fact_5700_trancl__power,axiom,
    ! [A: $tType,P: product_prod(A,A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),P),transitive_trancl(A,R2)))
    <=> ? [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),P),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N5),R2))) ) ) ).

% trancl_power
tff(fact_5701_card_Oeq__fold,axiom,
    ! [A: $tType,A6: set(A)] : aa(set(A),nat,finite_card(A),A6) = finite_fold(A,nat,aTP_Lamp_aas(A,fun(nat,nat)),zero_zero(nat),A6) ).

% card.eq_fold
tff(fact_5702_less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M2),N)),transitive_trancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N)) ) ).

% less_eq
tff(fact_5703_trancl__insert2,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : transitive_trancl(A,aa(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),A3),B2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_aat(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A3),B2),R)))) ).

% trancl_insert2
tff(fact_5704_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
     => ( transitive_rtrancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aak(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_aap(set(product_prod(A,A)),fun(nat,bool),R2)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_5705_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Ks: list(A)] : map_of(A,B,map(A,product_prod(A,B),aTP_Lamp_xh(fun(A,B),fun(A,product_prod(A,B)),F3),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F3),aa(list(A),set(A),set2(A),Ks)) ).

% map_of_map_restrict
tff(fact_5706_ntrancl__Suc,axiom,
    ! [A: $tType,N: nat,R2: set(product_prod(A,A))] : transitive_ntrancl(A,aa(nat,nat,suc,N),R2) = relcomp(A,A,A,transitive_ntrancl(A,N,R2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),id2(A)),R2)) ).

% ntrancl_Suc
tff(fact_5707_pair__in__Id__conv,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id2(A)))
    <=> ( A3 = B2 ) ) ).

% pair_in_Id_conv
tff(fact_5708_IdI,axiom,
    ! [A: $tType,A3: A] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),id2(A))) ).

% IdI
tff(fact_5709_trancl__rtrancl__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),transitive_rtrancl(A,R)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R))) ) ) ).

% trancl_rtrancl_trancl
tff(fact_5710_rtrancl__trancl__trancl,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A)),Z2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_trancl(A,R)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_trancl(A,R))) ) ) ).

% rtrancl_trancl_trancl
tff(fact_5711_rtrancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),transitive_rtrancl(A,R)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R))) ) ) ).

% rtrancl_into_trancl2
tff(fact_5712_rtrancl__into__trancl1,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),R))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R))) ) ) ).

% rtrancl_into_trancl1
tff(fact_5713_rtrancl__eq__or__trancl,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2)))
    <=> ( ( X = Y )
        | ( ( X != Y )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2))) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_5714_trancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R))) ) ).

% trancl_into_rtrancl
tff(fact_5715_tranclD2,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2)))
     => ? [Z: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),transitive_rtrancl(A,R2)))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z),Y)),R2)) ) ) ).

% tranclD2
tff(fact_5716_rtranclD,axiom,
    ! [A: $tType,A3: A,B2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R2)))
     => ( ( A3 = B2 )
        | ( ( A3 != B2 )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R2))) ) ) ) ).

% rtranclD
tff(fact_5717_tranclD,axiom,
    ! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R2)))
     => ? [Z: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),R2))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z),Y)),transitive_rtrancl(A,R2))) ) ) ).

% tranclD
tff(fact_5718_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R)))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),transitive_rtrancl(list(A),listrel1(A,R))))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),transitive_rtrancl(list(A),listrel1(A,R)))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_5719_rtrancl_Ocases,axiom,
    ! [A: $tType,A12: A,A23: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),A23)),transitive_rtrancl(A,R)))
     => ( ( A23 != A12 )
       => ~ ! [B4: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),B4)),transitive_rtrancl(A,R)))
             => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A23)),R)) ) ) ) ).

% rtrancl.cases
tff(fact_5720_rtrancl_Osimps,axiom,
    ! [A: $tType,A12: A,A23: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),A23)),transitive_rtrancl(A,R)))
    <=> ( ? [A7: A] :
            ( ( A12 = A7 )
            & ( A23 = A7 ) )
        | ? [A7: A,B5: A,C4: A] :
            ( ( A12 = A7 )
            & ( A23 = C4 )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B5)),transitive_rtrancl(A,R)))
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),C4)),R)) ) ) ) ).

% rtrancl.simps
tff(fact_5721_rtrancl_Ortrancl__refl,axiom,
    ! [A: $tType,A3: A,R: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),transitive_rtrancl(A,R))) ).

% rtrancl.rtrancl_refl
tff(fact_5722_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),R))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_rtrancl(A,R))) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_5723_rtranclE,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R)))
     => ( ( A3 != B2 )
       => ~ ! [Y3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y3)),transitive_rtrancl(A,R)))
             => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),B2)),R)) ) ) ) ).

% rtranclE
tff(fact_5724_rtrancl__trans,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A)),Z2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_rtrancl(A,R)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_rtrancl(A,R))) ) ) ).

% rtrancl_trans
tff(fact_5725_rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P2: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R)))
     => ( pp(aa(A,bool,P2,A3))
       => ( ! [Y3: A,Z: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y3)),transitive_rtrancl(A,R)))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R))
               => ( pp(aa(A,bool,P2,Y3))
                 => pp(aa(A,bool,P2,Z)) ) ) )
         => pp(aa(A,bool,P2,B2)) ) ) ) ).

% rtrancl_induct
tff(fact_5726_converse__rtranclE,axiom,
    ! [A: $tType,X: A,Z2: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_rtrancl(A,R)))
     => ( ( X != Z2 )
       => ~ ! [Y3: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R))
             => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2)),transitive_rtrancl(A,R))) ) ) ) ).

% converse_rtranclE
tff(fact_5727_converse__rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P2: fun(A,bool)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R)))
     => ( pp(aa(A,bool,P2,B2))
       => ( ! [Y3: A,Z: A] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z),B2)),transitive_rtrancl(A,R)))
               => ( pp(aa(A,bool,P2,Z))
                 => pp(aa(A,bool,P2,Y3)) ) ) )
         => pp(aa(A,bool,P2,A3)) ) ) ) ).

% converse_rtrancl_induct
tff(fact_5728_converse__rtrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),transitive_rtrancl(A,R)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_rtrancl(A,R))) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_5729_IdD,axiom,
    ! [A: $tType,A3: A,B2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id2(A)))
     => ( A3 = B2 ) ) ).

% IdD
tff(fact_5730_IdE,axiom,
    ! [A: $tType,P: product_prod(A,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),P),id2(A)))
     => ~ ! [X3: A] : P != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ).

% IdE
tff(fact_5731_rtrancl__Un__separatorE,axiom,
    ! [A: $tType,A3: A,B2: A,P2: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P2),Q))))
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3)),transitive_rtrancl(A,P2)))
           => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),Q))
             => ( X3 = Y3 ) ) )
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,P2))) ) ) ).

% rtrancl_Un_separatorE
tff(fact_5732_rtrancl__Un__separator__converseE,axiom,
    ! [A: $tType,A3: A,B2: A,P2: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P2),Q))))
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),B2)),transitive_rtrancl(A,P2)))
           => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3)),Q))
             => ( Y3 = X3 ) ) )
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,P2))) ) ) ).

% rtrancl_Un_separator_converseE
tff(fact_5733_converse__rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),P2: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By))
       => ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P2,Ax),Ay)) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_5734_converse__rtranclE2,axiom,
    ! [B: $tType,A: $tType,Xa2: A,Xb: B,Za: A,Zb: B,R: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))),transitive_rtrancl(product_prod(A,B),R)))
     => ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb) )
       => ~ ! [A5: A,B4: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),R))
             => ~ pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))),transitive_rtrancl(product_prod(A,B),R))) ) ) ) ).

% converse_rtranclE2
tff(fact_5735_rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),P2: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Ax),Ay))
       => ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
              ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),transitive_rtrancl(product_prod(A,B),R)))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,aa(product_prod(product_prod(A,B),product_prod(A,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),bool),member(product_prod(product_prod(A,B),product_prod(A,B))),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By)) ) ) ) ).

% rtrancl_induct2
tff(fact_5736_relpow_Osimps_I1_J,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2) = id2(A) ).

% relpow.simps(1)
tff(fact_5737_Id__def,axiom,
    ! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_aau(product_prod(A,A),bool)) ).

% Id_def
tff(fact_5738_ran__restrictD,axiom,
    ! [B: $tType,A: $tType,Y: A,M2: fun(B,option(A)),A6: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),ran(B,A,restrict_map(B,A,M2,A6))))
     => ? [X3: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
          & ( aa(B,option(A),M2,X3) = aa(A,option(A),some(A),Y) ) ) ) ).

% ran_restrictD
tff(fact_5739_rtrancl__listrel1__eq__len,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X),Y)),transitive_rtrancl(list(A),listrel1(A,R))))
     => ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_5740_pred__nat__trancl__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M2),N)),transitive_rtrancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_5741_reflcl__set__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X5: A,Xa: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),fequal(A)),X5),Xa))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),id2(A)))) ) ).

% reflcl_set_eq
tff(fact_5742_rtrancl__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : transitive_rtrancl(A,aa(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),A3),B2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_aav(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A3),B2),R)))) ).

% rtrancl_insert
tff(fact_5743_trancl__insert,axiom,
    ! [A: $tType,Y: A,X: A,R: set(product_prod(A,A))] : transitive_trancl(A,aa(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),Y),X)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_aav(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Y),X),R)))) ).

% trancl_insert
tff(fact_5744_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),D5: set(A),M2: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D5))
       => ( restrict_map(A,B,map_upds(A,B,M2,Xs,Ys),D5) = map_upds(A,B,restrict_map(A,B,M2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D5),aa(list(A),set(A),set2(A),Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_5745_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_lw(A,fun(A,bool))) = ord_min(A) ) ) ).

% dual_max
tff(fact_5746_total__on__singleton,axiom,
    ! [A: $tType,X: A] : total_on(A,aa(set(A),set(A),insert(A,X),bot_bot(set(A))),aa(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),X),X)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_5747_fun__upds__append__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),M2: fun(A,option(B)),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,M2,append(A,Xs,Zs),Ys) = map_upds(A,B,M2,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_5748_fun__upds__append2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),M2: fun(A,option(B)),Zs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,M2,Xs,append(B,Ys,Zs)) = map_upds(A,B,M2,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_5749_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I2: nat,M2: fun(A,option(B)),Ys: list(B),Y: B] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2))
     => ( map_upds(A,B,M2,Xs,list_update(B,Ys,I2,Y)) = map_upds(A,B,M2,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_5750_total__onI,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
      ( ! [X3: A,Y3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A6))
           => ( ( X3 != Y3 )
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R))
                | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3)),R)) ) ) ) )
     => total_on(A,A6,R) ) ).

% total_onI
tff(fact_5751_total__on__def,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
      ( total_on(A,A6,R)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A6))
             => ( ( X4 != Xa3 )
               => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R))
                  | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X4)),R)) ) ) ) ) ) ).

% total_on_def
tff(fact_5752_ord_Omax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : max(A,Less_eq) = max(A,Less_eq) ).

% ord.max.cong
tff(fact_5753_ord_Omax__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A3),B2) = B2 ) )
      & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A3),B2) = A3 ) ) ) ).

% ord.max_def
tff(fact_5754_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),M2: fun(A,option(B)),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)))
     => ( map_upds(A,B,M2,append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),Ys) = fun_upd(A,option(B),map_upds(A,B,M2,Xs,Ys),X,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_5755_bezw__aux,axiom,
    ! [X: nat,Y: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),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_5756_Field__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : field2(A,aa(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),A3),B2)),R)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))),field2(A,R)) ).

% Field_insert
tff(fact_5757_gcd__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% gcd_eq_0_iff
tff(fact_5758_gcd__0__left__nat,axiom,
    ! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),X) = X ).

% gcd_0_left_nat
tff(fact_5759_gcd__0__nat,axiom,
    ! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),zero_zero(nat)) = X ).

% gcd_0_nat
tff(fact_5760_gcd__nat_Oright__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),zero_zero(nat)) = A3 ).

% gcd_nat.right_neutral
tff(fact_5761_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_5762_gcd__nat_Oleft__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),A3) = A3 ).

% gcd_nat.left_neutral
tff(fact_5763_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) = zero_zero(nat) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_5764_gcd__Suc__0,axiom,
    ! [M2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% gcd_Suc_0
tff(fact_5765_gcd__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),N)))
    <=> ( ( M2 != zero_zero(nat) )
        | ( N != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_5766_image__map__upd,axiom,
    ! [B: $tType,A: $tType,X: A,A6: set(A),M2: fun(A,option(B)),Y: B] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
     => ( image2(A,option(B),fun_upd(A,option(B),M2,X,aa(B,option(B),some(B),Y)),A6) = image2(A,option(B),M2,A6) ) ) ).

% image_map_upd
tff(fact_5767_map__upds__Cons,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B)),A3: A,As2: list(A),B2: B,Bs: list(B)] : map_upds(A,B,M2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs)) = map_upds(A,B,fun_upd(A,option(B),M2,A3,aa(B,option(B),some(B),B2)),As2,Bs) ).

% map_upds_Cons
tff(fact_5768_map__upds__twist,axiom,
    ! [A: $tType,B: $tType,A3: A,As2: list(A),M2: fun(A,option(B)),B2: B,Bs: list(B)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),aa(list(A),set(A),set2(A),As2)))
     => ( map_upds(A,B,fun_upd(A,option(B),M2,A3,aa(B,option(B),some(B),B2)),As2,Bs) = fun_upd(A,option(B),map_upds(A,B,M2,As2,Bs),A3,aa(B,option(B),some(B),B2)) ) ) ).

% map_upds_twist
tff(fact_5769_ran__map__upd,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),M2,A3) = none(A) )
     => ( ran(B,A,fun_upd(B,option(A),M2,A3,aa(A,option(A),some(A),B2))) = aa(set(A),set(A),insert(A,B2),ran(B,A,M2)) ) ) ).

% ran_map_upd
tff(fact_5770_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),X: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),M2,X,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) = restrict_map(A,B,M2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_5771_map__upd__Some__unfold,axiom,
    ! [B: $tType,A: $tType,M2: fun(B,option(A)),A3: B,B2: A,X: B,Y: A] :
      ( ( aa(B,option(A),fun_upd(B,option(A),M2,A3,aa(A,option(A),some(A),B2)),X) = aa(A,option(A),some(A),Y) )
    <=> ( ( ( X = A3 )
          & ( B2 = Y ) )
        | ( ( X != A3 )
          & ( aa(B,option(A),M2,X) = aa(A,option(A),some(A),Y) ) ) ) ) ).

% map_upd_Some_unfold
tff(fact_5772_map__upd__triv,axiom,
    ! [A: $tType,B: $tType,T2: fun(B,option(A)),K2: B,X: A] :
      ( ( aa(B,option(A),T2,K2) = aa(A,option(A),some(A),X) )
     => ( fun_upd(B,option(A),T2,K2,aa(A,option(A),some(A),X)) = T2 ) ) ).

% map_upd_triv
tff(fact_5773_map__upd__eqD1,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B)),A3: A,X: B,N: fun(A,option(B)),Y: B] :
      ( ( fun_upd(A,option(B),M2,A3,aa(B,option(B),some(B),X)) = fun_upd(A,option(B),N,A3,aa(B,option(B),some(B),Y)) )
     => ( X = Y ) ) ).

% map_upd_eqD1
tff(fact_5774_map__upd__nonempty,axiom,
    ! [A: $tType,B: $tType,T2: fun(A,option(B)),K2: A,X: B] :
      ~ ! [X3: A] : aa(A,option(B),fun_upd(A,option(B),T2,K2,aa(B,option(B),some(B),X)),X3) = none(B) ).

% map_upd_nonempty
tff(fact_5775_gcd__diff2__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),N) ) ) ).

% gcd_diff2_nat
tff(fact_5776_gcd__diff1__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M2),N)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),N) ) ) ).

% gcd_diff1_nat
tff(fact_5777_gcd__le1__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),A3)) ) ).

% gcd_le1_nat
tff(fact_5778_gcd__le2__nat,axiom,
    ! [B2: nat,A3: nat] :
      ( ( B2 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),B2)) ) ).

% gcd_le2_nat
tff(fact_5779_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,K2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),K2),B2))) ) ).

% gcd_dvd_prod
tff(fact_5780_gcd__mult__distrib__nat,axiom,
    ! [K2: nat,M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),N)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),M2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),N)) ).

% gcd_mult_distrib_nat
tff(fact_5781_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M2: A,K2: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K2),M2)),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M2),N) ) ).

% gcd_add_mult
tff(fact_5782_FieldI1,axiom,
    ! [A: $tType,I2: A,J2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2)),R2))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),field2(A,R2))) ) ).

% FieldI1
tff(fact_5783_FieldI2,axiom,
    ! [A: $tType,I2: A,J2: A,R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2)),R2))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),field2(A,R2))) ) ).

% FieldI2
tff(fact_5784_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),modulo_modulo(nat,X,Y)) ) ) ).

% gcd_non_0_nat
tff(fact_5785_gcd__nat_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = X ) )
      & ( ( Y != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),modulo_modulo(nat,X,Y)) ) ) ) ).

% gcd_nat.simps
tff(fact_5786_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa2) = Y )
     => ( ( ( Xa2 = zero_zero(nat) )
         => ( Y = X ) )
        & ( ( Xa2 != zero_zero(nat) )
         => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa2),modulo_modulo(nat,X,Xa2)) ) ) ) ) ).

% gcd_nat.elims
tff(fact_5787_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C3) ) ) ) ).

% gcd_mult_unit2
tff(fact_5788_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C3) ) ) ) ).

% gcd_mult_unit1
tff(fact_5789_bezout__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [X3: nat,Y3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)) ) ).

% bezout_nat
tff(fact_5790_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A3: nat] :
    ? [X3: nat,Y3: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) )
      | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_5791_Field__natLeq__on,axiom,
    ! [N: nat] : field2(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aaw(nat,fun(nat,fun(nat,bool)),N)))) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_bt(nat,fun(nat,bool)),N)) ).

% Field_natLeq_on
tff(fact_5792_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_aax(nat,fun(nat,bool))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_5793_finite__range__updI,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),A3: B,B2: A] :
      ( pp(aa(set(option(A)),bool,finite_finite2(option(A)),image2(B,option(A),F3,top_top(set(B)))))
     => pp(aa(set(option(A)),bool,finite_finite2(option(A)),image2(B,option(A),fun_upd(B,option(A),F3,A3,aa(A,option(A),some(A),B2)),top_top(set(B))))) ) ).

% finite_range_updI
tff(fact_5794_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_aay(nat,fun(nat,fun(nat,bool)),N),M2))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_5795_Gcd__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite2(A),A6))
           => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A6) = finite_fold(A,A,gcd_gcd(A),zero_zero(A),A6) ) )
          & ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
           => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A6) = one_one(A) ) ) ) ) ).

% Gcd_fin.eq_fold
tff(fact_5796_map__of__zip__upd,axiom,
    ! [A: $tType,B: $tType,Ys: list(B),Xs: list(A),Zs: list(B),X: A,Y: B,Z2: B] :
      ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(A),nat,size_size(list(A)),Xs) )
     => ( ( aa(list(B),nat,size_size(list(B)),Zs) = aa(list(A),nat,size_size(list(A)),Xs) )
       => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
         => ( ( fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X,aa(B,option(B),some(B),Y)) = fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Zs)),X,aa(B,option(B),some(B),Z2)) )
           => ( map_of(A,B,zip(A,B,Xs,Ys)) = map_of(A,B,zip(A,B,Xs,Zs)) ) ) ) ) ) ).

% map_of_zip_upd
tff(fact_5797_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,X: A,Ys: list(B),Xs: list(A),F3: fun(A,option(B)),Y: B] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F3,X,aa(B,option(B),some(B),Y)),Xs,Ys) = map_upds(A,B,F3,Xs,Ys) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F3,X,aa(B,option(B),some(B),Y)),Xs,Ys) = fun_upd(A,option(B),map_upds(A,B,F3,Xs,Ys),X,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_5798_map__of_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType,P: product_prod(A,B),Ps: list(product_prod(A,B))] : map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),P),Ps)) = fun_upd(A,option(B),map_of(A,B,Ps),aa(product_prod(A,B),A,product_fst(A,B),P),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),P))) ).

% map_of.simps(2)
tff(fact_5799_Total__subset__Id,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( total_on(A,field2(A,R),R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),id2(A)))
       => ( ( R = bot_bot(set(product_prod(A,A))) )
          | ? [A5: A] : R = aa(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),A5),A5)),bot_bot(set(product_prod(A,A)))) ) ) ) ).

% Total_subset_Id
tff(fact_5800_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa2) = Y )
     => ( accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2))
       => ~ ( ( ( ( Xa2 = zero_zero(nat) )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero(nat) )
               => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa2),modulo_modulo(nat,X,Xa2)) ) ) )
           => ~ accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2)) ) ) ) ).

% gcd_nat.pelims
tff(fact_5801_ran__map__upd__Some,axiom,
    ! [B: $tType,A: $tType,M2: fun(B,option(A)),X: B,Y: A,Z2: A] :
      ( ( aa(B,option(A),M2,X) = aa(A,option(A),some(A),Y) )
     => ( inj_on(B,option(A),M2,dom(B,A,M2))
       => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),ran(B,A,M2)))
         => ( ran(B,A,fun_upd(B,option(A),M2,X,aa(A,option(A),some(A),Z2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),ran(B,A,M2)),aa(set(A),set(A),insert(A,Y),bot_bot(set(A))))),aa(set(A),set(A),insert(A,Z2),bot_bot(set(A)))) ) ) ) ) ).

% ran_map_upd_Some
tff(fact_5802_dom__const,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : dom(A,B,aTP_Lamp_aaz(fun(A,B),fun(A,option(B)),F3)) = top_top(set(A)) ).

% dom_const
tff(fact_5803_dom__map__of__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( dom(A,B,map_of(A,B,zip(A,B,Xs,Ys))) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% dom_map_of_zip
tff(fact_5804_dom__map__upds,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),Xs: list(A),Ys: list(B)] : dom(A,B,map_upds(A,B,M2,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))),dom(A,B,M2)) ).

% dom_map_upds
tff(fact_5805_gcd__mult__distrib__int,axiom,
    ! [K2: int,M2: int,N: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),K2)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M2),N)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),M2)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),N)) ).

% gcd_mult_distrib_int
tff(fact_5806_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U4: int,V2: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),U4),X)),aa(int,int,aa(int,fun(int,int),times_times(int),V2),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).

% bezout_int
tff(fact_5807_domD,axiom,
    ! [A: $tType,B: $tType,A3: A,M2: fun(A,option(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),dom(A,B,M2)))
     => ? [B4: B] : aa(A,option(B),M2,A3) = aa(B,option(B),some(B),B4) ) ).

% domD
tff(fact_5808_domI,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A)),A3: B,B2: A] :
      ( ( aa(B,option(A),M2,A3) = aa(A,option(A),some(A),B2) )
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),dom(B,A,M2))) ) ).

% domI
tff(fact_5809_insert__dom,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),X: B,Y: A] :
      ( ( aa(B,option(A),F3,X) = aa(A,option(A),some(A),Y) )
     => ( aa(set(B),set(B),insert(B,X),dom(B,A,F3)) = dom(B,A,F3) ) ) ).

% insert_dom
tff(fact_5810_finite__Map__induct,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),P2: fun(fun(A,option(B)),bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M2)))
     => ( pp(aa(fun(A,option(B)),bool,P2,aTP_Lamp_aba(A,option(B))))
       => ( ! [K: A,V2: B,M: fun(A,option(B))] :
              ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M)))
             => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),K),dom(A,B,M)))
               => ( pp(aa(fun(A,option(B)),bool,P2,M))
                 => pp(aa(fun(A,option(B)),bool,P2,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V2)))) ) ) )
         => pp(aa(fun(A,option(B)),bool,P2,M2)) ) ) ) ).

% finite_Map_induct
tff(fact_5811_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),X: A] :
      ( ( dom(A,B,F3) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
    <=> ? [V4: B] : F3 = fun_upd(A,option(B),aTP_Lamp_aba(A,option(B)),X,aa(B,option(B),some(B),V4)) ) ).

% dom_eq_singleton_conv
tff(fact_5812_map__of__map__keys,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),M2: fun(A,option(B))] :
      ( ( aa(list(A),set(A),set2(A),Xs) = dom(A,B,M2) )
     => ( map_of(A,B,map(A,product_prod(A,B),aTP_Lamp_abb(fun(A,option(B)),fun(A,product_prod(A,B)),M2),Xs)) = M2 ) ) ).

% map_of_map_keys
tff(fact_5813_UnderS__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A6: set(A)] : order_UnderS(A,R,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_abc(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A6)) ).

% UnderS_def
tff(fact_5814_Under__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A6: set(A)] : order_Under(A,R,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_abd(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A6)) ).

% Under_def
tff(fact_5815_Above__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A6: set(A)] : order_Above(A,R,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_abe(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A6)) ).

% Above_def
tff(fact_5816_cofinal__def,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
      ( bNF_Ca7293521722713021262ofinal(A,A6,R)
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),field2(A,R)))
         => ? [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A6))
              & ( X4 != Xa3 )
              & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R)) ) ) ) ).

% cofinal_def
tff(fact_5817_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B)),K2: A,V3: B] : graph(A,B,fun_upd(A,option(B),M2,K2,aa(B,option(B),some(B),V3))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V3)),graph(A,B,fun_upd(A,option(B),M2,K2,none(B)))) ).

% graph_map_upd
tff(fact_5818_linear__order__on__singleton,axiom,
    ! [A: $tType,X: A] : order_679001287576687338der_on(A,aa(set(A),set(A),insert(A,X),bot_bot(set(A))),aa(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),X),X)),bot_bot(set(product_prod(A,A))))) ).

% linear_order_on_singleton
tff(fact_5819_in__graphD,axiom,
    ! [A: $tType,B: $tType,K2: A,V3: B,M2: fun(A,option(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V3)),graph(A,B,M2)))
     => ( aa(A,option(B),M2,K2) = aa(B,option(B),some(B),V3) ) ) ).

% in_graphD
tff(fact_5820_in__graphI,axiom,
    ! [A: $tType,B: $tType,M2: fun(B,option(A)),K2: B,V3: A] :
      ( ( aa(B,option(A),M2,K2) = aa(A,option(A),some(A),V3) )
     => pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K2),V3)),graph(B,A,M2))) ) ).

% in_graphI
tff(fact_5821_graph__restrictD_I1_J,axiom,
    ! [B: $tType,A: $tType,K2: A,V3: B,M2: fun(A,option(B)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V3)),graph(A,B,restrict_map(A,B,M2,A6))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),K2),A6)) ) ).

% graph_restrictD(1)
tff(fact_5822_graph__restrictD_I2_J,axiom,
    ! [A: $tType,B: $tType,K2: A,V3: B,M2: fun(A,option(B)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V3)),graph(A,B,restrict_map(A,B,M2,A6))))
     => ( aa(A,option(B),M2,K2) = aa(B,option(B),some(B),V3) ) ) ).

% graph_restrictD(2)
tff(fact_5823_graph__def,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] : graph(A,B,M2) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_abf(fun(A,option(B)),fun(product_prod(A,B),bool),M2)) ).

% graph_def
tff(fact_5824_Linear__order__in__diff__Id,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R)))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
          <=> ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_5825_graph__eq__to__snd__dom,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B))] : graph(A,B,M2) = image2(A,product_prod(A,B),aTP_Lamp_abb(fun(A,option(B)),fun(A,product_prod(A,B)),M2),dom(A,B,M2)) ).

% graph_eq_to_snd_dom
tff(fact_5826_map__upds__fold__map__upd,axiom,
    ! [A: $tType,B: $tType,M2: fun(A,option(B)),Ks: list(A),Vs: list(B)] : map_upds(A,B,M2,Ks,Vs) = foldl(fun(A,option(B)),product_prod(A,B),aTP_Lamp_abh(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),M2,zip(A,B,Ks,Vs)) ).

% map_upds_fold_map_upd
tff(fact_5827_increasing__Bseq__subseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),G3: fun(nat,nat)] :
          ( ! [X3: nat,Y3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Y3))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,X3))),real_V7770717601297561774m_norm(A,aa(nat,A,F3,Y3)))) )
         => ( order_strict_mono(nat,nat,G3)
           => ( bfun(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_abi(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),F3),G3),at_top(nat))
            <=> bfun(nat,A,F3,at_top(nat)) ) ) ) ) ).

% increasing_Bseq_subseq_iff
tff(fact_5828_butlast__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),butlast(A),take(A,N,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_5829_length__butlast,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),butlast(A),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).

% length_butlast
tff(fact_5830_strict__mono__imp__increasing,axiom,
    ! [F3: fun(nat,nat),N: nat] :
      ( order_strict_mono(nat,nat,F3)
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,F3,N))) ) ).

% strict_mono_imp_increasing
tff(fact_5831_strict__mono__less__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F3)
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% strict_mono_less_eq
tff(fact_5832_strict__mono__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [R: fun(A,B),M2: A,N: A] :
          ( order_strict_mono(A,B,R)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),N))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,R,M2)),aa(A,B,R,N))) ) ) ) ).

% strict_mono_leD
tff(fact_5833_strict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ).

% strict_monoD
tff(fact_5834_strict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X3)),aa(A,B,F3,Y3))) )
         => order_strict_mono(A,B,F3) ) ) ).

% strict_monoI
tff(fact_5835_strict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_strict_mono(A,B,F3)
        <=> ! [X4: A,Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X4)),aa(A,B,F3,Y5))) ) ) ) ).

% strict_mono_def
tff(fact_5836_strict__mono__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F3)
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% strict_mono_less
tff(fact_5837_strict__mono__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F3)
         => ( ( aa(A,B,F3,X) = aa(A,B,F3,Y) )
          <=> ( X = Y ) ) ) ) ).

% strict_mono_eq
tff(fact_5838_strict__mono__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_strict_mono(A,B,F3)
         => pp(aa(fun(A,B),bool,order_mono(A,B),F3)) ) ) ).

% strict_mono_mono
tff(fact_5839_strict__mono__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( order_strict_mono(nat,A,F3)
        <=> ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N5)),aa(nat,A,F3,aa(nat,nat,suc,N5)))) ) ) ).

% strict_mono_Suc_iff
tff(fact_5840_nth__butlast,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),butlast(A),Xs))))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),butlast(A),Xs)),N) = aa(nat,A,nth(A,Xs),N) ) ) ).

% nth_butlast
tff(fact_5841_sorted__butlast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),butlast(A),Xs)) ) ) ) ).

% sorted_butlast
tff(fact_5842_take__butlast,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( take(A,N,aa(list(A),list(A),butlast(A),Xs)) = take(A,N,Xs) ) ) ).

% take_butlast
tff(fact_5843_butlast__power,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(nat,fun(fun(list(A),list(A)),fun(list(A),list(A))),compow(fun(list(A),list(A))),N),butlast(A)),Xs) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N),Xs) ).

% butlast_power
tff(fact_5844_butlast__conv__take,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),butlast(A),Xs) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),Xs) ).

% butlast_conv_take
tff(fact_5845_butlast__list__update,axiom,
    ! [A: $tType,K2: nat,Xs: list(A),X: A] :
      ( ( ( K2 = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
       => ( aa(list(A),list(A),butlast(A),list_update(A,Xs,K2,X)) = aa(list(A),list(A),butlast(A),Xs) ) )
      & ( ( K2 != aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
       => ( aa(list(A),list(A),butlast(A),list_update(A,Xs,K2,X)) = list_update(A,aa(list(A),list(A),butlast(A),Xs),K2,X) ) ) ) ).

% butlast_list_update
tff(fact_5846_summable__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G3: fun(nat,nat),F3: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G3)
         => ( ! [N3: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N3),image2(nat,nat,G3,top_top(set(nat)))))
               => ( aa(nat,A,F3,N3) = zero_zero(A) ) )
           => ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_abj(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G3),F3))
            <=> summable(A,F3) ) ) ) ) ).

% summable_mono_reindex
tff(fact_5847_sums__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G3: fun(nat,nat),F3: fun(nat,A),C3: A] :
          ( order_strict_mono(nat,nat,G3)
         => ( ! [N3: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N3),image2(nat,nat,G3,top_top(set(nat)))))
               => ( aa(nat,A,F3,N3) = zero_zero(A) ) )
           => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_abj(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G3),F3),C3)
            <=> sums(A,F3,C3) ) ) ) ) ).

% sums_mono_reindex
tff(fact_5848_suminf__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [G3: fun(nat,nat),F3: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G3)
         => ( ! [N3: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N3),image2(nat,nat,G3,top_top(set(nat)))))
               => ( aa(nat,A,F3,N3) = zero_zero(A) ) )
           => ( suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_abk(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G3),F3)) = suminf(A,F3) ) ) ) ) ).

% suminf_mono_reindex
tff(fact_5849_Linear__order__wf__diff__Id,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))
      <=> ! [A11: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A11),field2(A,R)))
           => ( ( A11 != bot_bot(set(A)) )
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A11))
                  & ! [Xa3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A11))
                     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R)) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_5850_bsqr__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R) = aa(fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),set(product_prod(product_prod(A,A),product_prod(A,A))),collect(product_prod(product_prod(A,A),product_prod(A,A))),aa(fun(product_prod(A,A),fun(product_prod(A,A),bool)),fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),product_case_prod(product_prod(A,A),product_prod(A,A),bool),aa(fun(A,fun(A,fun(product_prod(A,A),bool))),fun(product_prod(A,A),fun(product_prod(A,A),bool)),product_case_prod(A,A,fun(product_prod(A,A),bool)),aTP_Lamp_abm(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),R)))) ).

% bsqr_def
tff(fact_5851_DeMoivre2,axiom,
    ! [R: real,A3: real,N: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),rcis(R,A3)),N) = rcis(aa(nat,real,aa(real,fun(nat,real),power_power(real),R),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),A3)) ).

% DeMoivre2
tff(fact_5852_wf__insert,axiom,
    ! [A: $tType,Y: A,X: A,R: set(product_prod(A,A))] :
      ( wf(A,aa(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),Y),X)),R))
    <=> ( wf(A,R)
        & ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R))) ) ) ).

% wf_insert
tff(fact_5853_Re__rcis,axiom,
    ! [R: real,A3: real] : re(rcis(R,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),R),cos(real,A3)) ).

% Re_rcis
tff(fact_5854_Im__rcis,axiom,
    ! [R: real,A3: real] : im(rcis(R,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),R),sin(real,A3)) ).

% Im_rcis
tff(fact_5855_wf__iff__no__infinite__down__chain,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ~ ? [F5: fun(nat,A)] :
          ! [I: nat] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F5,aa(nat,nat,suc,I))),aa(nat,A,F5,I))),R)) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_5856_wf__no__infinite__down__chainE,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),F3: fun(nat,A)] :
      ( wf(A,R)
     => ~ ! [K: nat] : pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F3,aa(nat,nat,suc,K))),aa(nat,A,F3,K))),R)) ) ).

% wf_no_infinite_down_chainE
tff(fact_5857_wf__induct__rule,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P2: fun(A,bool),A3: A] :
      ( wf(A,R)
     => ( ! [X3: A] :
            ( ! [Y4: A] :
                ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R))
               => pp(aa(A,bool,P2,Y4)) )
           => pp(aa(A,bool,P2,X3)) )
       => pp(aa(A,bool,P2,A3)) ) ) ).

% wf_induct_rule
tff(fact_5858_wf__eq__minimal,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ! [Q6: set(A)] :
          ( ? [X4: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Q6))
         => ? [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Q6))
              & ! [Y5: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X4)),R))
                 => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y5),Q6)) ) ) ) ) ).

% wf_eq_minimal
tff(fact_5859_wf__not__refl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] :
      ( wf(A,R)
     => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R)) ) ).

% wf_not_refl
tff(fact_5860_wf__not__sym,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,X: A] :
      ( wf(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X)),R))
       => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3)),R)) ) ) ).

% wf_not_sym
tff(fact_5861_wf__irrefl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] :
      ( wf(A,R)
     => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R)) ) ).

% wf_irrefl
tff(fact_5862_wf__induct,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P2: fun(A,bool),A3: A] :
      ( wf(A,R)
     => ( ! [X3: A] :
            ( ! [Y4: A] :
                ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R))
               => pp(aa(A,bool,P2,Y4)) )
           => pp(aa(A,bool,P2,X3)) )
       => pp(aa(A,bool,P2,A3)) ) ) ).

% wf_induct
tff(fact_5863_wf__asym,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,X: A] :
      ( wf(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X)),R))
       => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3)),R)) ) ) ).

% wf_asym
tff(fact_5864_wfUNIVI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [P5: fun(A,bool),X3: A] :
          ( ! [Xa: A] :
              ( ! [Y3: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Xa)),R))
                 => pp(aa(A,bool,P5,Y3)) )
             => pp(aa(A,bool,P5,Xa)) )
         => pp(aa(A,bool,P5,X3)) )
     => wf(A,R) ) ).

% wfUNIVI
tff(fact_5865_wfI__min,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X3: A,Q7: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),Q7))
         => ? [Xa: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),Q7))
              & ! [Y3: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Xa)),R2))
                 => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),Q7)) ) ) )
     => wf(A,R2) ) ).

% wfI_min
tff(fact_5866_wfE__min,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,Q: set(A)] :
      ( wf(A,R2)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),Q))
       => ~ ! [Z: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),Q))
             => ~ ! [Y4: A] :
                    ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),R2))
                   => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),Q)) ) ) ) ) ).

% wfE_min
tff(fact_5867_wf__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ! [P6: fun(A,bool)] :
          ( ! [X4: A] :
              ( ! [Y5: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X4)),R))
                 => pp(aa(A,bool,P6,Y5)) )
             => pp(aa(A,bool,P6,X4)) )
         => ! [X_13: A] : pp(aa(A,bool,P6,X_13)) ) ) ).

% wf_def
tff(fact_5868_wfE__min_H,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Q: set(A)] :
      ( wf(A,R2)
     => ( ( Q != bot_bot(set(A)) )
       => ~ ! [Z: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),Q))
             => ~ ! [Y4: A] :
                    ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),R2))
                   => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),Q)) ) ) ) ) ).

% wfE_min'
tff(fact_5869_wf__bounded__measure,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Ub: fun(A,nat),F3: fun(A,nat)] :
      ( ! [A5: A,B4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5)),R))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Ub,B4)),aa(A,nat,Ub,A5)))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,B4)),aa(A,nat,Ub,A5)))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,A5)),aa(A,nat,F3,B4))) ) )
     => wf(A,R) ) ).

% wf_bounded_measure
tff(fact_5870_wf__linord__ex__has__least,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),P2: fun(B,bool),K2: B,M2: fun(B,A)] :
      ( wf(A,R)
     => ( ! [X3: A,Y3: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),transitive_trancl(A,R)))
          <=> ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3)),transitive_rtrancl(A,R))) )
       => ( pp(aa(B,bool,P2,K2))
         => ? [X3: B] :
              ( pp(aa(B,bool,P2,X3))
              & ! [Y4: B] :
                  ( pp(aa(B,bool,P2,Y4))
                 => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,M2,X3)),aa(B,A,M2,Y4))),transitive_rtrancl(A,R))) ) ) ) ) ) ).

% wf_linord_ex_has_least
tff(fact_5871_wf__eq__minimal2,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ! [A11: set(A)] :
          ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A11),field2(A,R)))
            & ( A11 != bot_bot(set(A)) ) )
         => ? [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A11))
              & ! [Xa3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A11))
                 => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X4)),R)) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_5872_wf__bounded__set,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),Ub: fun(A,set(B)),F3: fun(A,set(B))] :
      ( ! [A5: A,B4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5)),R))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),Ub,A5)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Ub,B4)),aa(A,set(B),Ub,A5)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F3,B4)),aa(A,set(B),Ub,A5)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(A,set(B),F3,A5)),aa(A,set(B),F3,B4))) ) )
     => wf(A,R) ) ).

% wf_bounded_set
tff(fact_5873_rcis__mult,axiom,
    ! [R1: real,A3: real,R22: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),rcis(R1,A3)),rcis(R22,B2)) = rcis(aa(real,real,aa(real,fun(real,real),times_times(real),R1),R22),aa(real,real,aa(real,fun(real,real),plus_plus(real),A3),B2)) ).

% rcis_mult
tff(fact_5874_rcis__def,axiom,
    ! [R: real,A3: real] : rcis(R,A3) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R)),cis(A3)) ).

% rcis_def
tff(fact_5875_dependent__wf__choice,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P2: fun(fun(A,B),fun(A,fun(B,bool)))] :
      ( wf(A,R2)
     => ( ! [F2: fun(A,B),G2: fun(A,B),X3: A,R3: B] :
            ( ! [Z4: A] :
                ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),X3)),R2))
               => ( aa(A,B,F2,Z4) = aa(A,B,G2,Z4) ) )
           => ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F2),X3),R3))
            <=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,G2),X3),R3)) ) )
       => ( ! [X3: A,F2: fun(A,B)] :
              ( ! [Y4: A] :
                  ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R2))
                 => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F2),Y4),aa(A,B,F2,Y4))) )
             => ? [X_1: B] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F2),X3),X_1)) )
         => ? [F2: fun(A,B)] :
            ! [X5: A] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F2),X5),aa(A,B,F2,X5))) ) ) ) ).

% dependent_wf_choice
tff(fact_5876_lexn_Osimps_I2_J,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),N: nat] : lexn(A,R,aa(nat,nat,suc,N)) = aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),inf_inf(set(product_prod(list(A),list(A)))),image2(product_prod(product_prod(A,list(A)),product_prod(A,list(A))),product_prod(list(A),list(A)),product_map_prod(product_prod(A,list(A)),list(A),product_prod(A,list(A)),list(A),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A)),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A))),lex_prod(A,list(A),R,lexn(A,R,N)))),aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_abn(nat,fun(list(A),fun(list(A),bool)),N)))) ).

% lexn.simps(2)
tff(fact_5877_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F3: fun(nat,fun(A,A)),A3: nat,B2: nat,Acc2: A] :
      ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A3),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc2))))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
         => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = Acc2 ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
         => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = set_fo6178422350223883121st_nat(A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F3,A3),Acc2)) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_5878_map__prod__ident,axiom,
    ! [B: $tType,A: $tType,X5: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_abo(A,A),aTP_Lamp_abp(B,B)),X5) = X5 ).

% map_prod_ident
tff(fact_5879_map__prod__simp,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(C,A),G3: fun(D,B),A3: C,B2: D] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F3,G3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A3),B2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,A3)),aa(D,B,G3,B2)) ).

% map_prod_simp
tff(fact_5880_fst__map__prod,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(C,A),G3: fun(D,B),X: product_prod(C,D)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F3,G3),X)) = aa(C,A,F3,aa(product_prod(C,D),C,product_fst(C,D),X)) ).

% fst_map_prod
tff(fact_5881_snd__map__prod,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(C,B),G3: fun(D,A),X: product_prod(C,D)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,D),product_prod(B,A),product_map_prod(C,B,D,A,F3,G3),X)) = aa(D,A,G3,aa(product_prod(C,D),D,product_snd(C,D),X)) ).

% snd_map_prod
tff(fact_5882_map__prod__imageI,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R2: set(product_prod(A,B)),F3: fun(A,C),G3: fun(B,D)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R2))
     => pp(aa(set(product_prod(C,D)),bool,aa(product_prod(C,D),fun(set(product_prod(C,D)),bool),member(product_prod(C,D)),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,F3,A3)),aa(B,D,G3,B2))),image2(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F3,G3),R2))) ) ).

% map_prod_imageI
tff(fact_5883_fst__comp__map__prod,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,F3: fun(A,C),G3: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),C),comp(product_prod(C,D),C,product_prod(A,B),product_fst(C,D)),product_map_prod(A,C,B,D,F3,G3)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F3),product_fst(A,B)) ).

% fst_comp_map_prod
tff(fact_5884_snd__comp__map__prod,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,F3: fun(A,D),G3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(D,C)),fun(product_prod(A,B),C),comp(product_prod(D,C),C,product_prod(A,B),product_snd(D,C)),product_map_prod(A,D,B,C,F3,G3)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),comp(B,C,product_prod(A,B),G3),product_snd(A,B)) ).

% snd_comp_map_prod
tff(fact_5885_map__prod__compose,axiom,
    ! [D: $tType,C: $tType,A: $tType,E: $tType,F: $tType,B: $tType,F1: fun(E,C),F22: fun(A,E),G1: fun(F,D),G22: fun(B,F)] : product_map_prod(A,C,B,D,aa(fun(A,E),fun(A,C),comp(E,C,A,F1),F22),aa(fun(B,F),fun(B,D),comp(F,D,B,G1),G22)) = aa(fun(product_prod(A,B),product_prod(E,F)),fun(product_prod(A,B),product_prod(C,D)),comp(product_prod(E,F),product_prod(C,D),product_prod(A,B),product_map_prod(E,C,F,D,F1,G1)),product_map_prod(A,E,B,F,F22,G22)) ).

% map_prod_compose
tff(fact_5886_map__prod_Ocompositionality,axiom,
    ! [D: $tType,F: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F3: fun(C,E),G3: fun(D,F),H: fun(A,C),I2: fun(B,D),Prod: product_prod(A,B)] : aa(product_prod(C,D),product_prod(E,F),product_map_prod(C,E,D,F,F3,G3),aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,H,I2),Prod)) = aa(product_prod(A,B),product_prod(E,F),product_map_prod(A,E,B,F,aa(fun(A,C),fun(A,E),comp(C,E,A,F3),H),aa(fun(B,D),fun(B,F),comp(D,F,B,G3),I2)),Prod) ).

% map_prod.compositionality
tff(fact_5887_map__prod_Ocomp,axiom,
    ! [A: $tType,C: $tType,E: $tType,F: $tType,D: $tType,B: $tType,F3: fun(C,E),G3: fun(D,F),H: fun(A,C),I2: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),product_prod(E,F)),comp(product_prod(C,D),product_prod(E,F),product_prod(A,B),product_map_prod(C,E,D,F,F3,G3)),product_map_prod(A,C,B,D,H,I2)) = product_map_prod(A,E,B,F,aa(fun(A,C),fun(A,E),comp(C,E,A,F3),H),aa(fun(B,D),fun(B,F),comp(D,F,B,G3),I2)) ).

% map_prod.comp
tff(fact_5888_prod__fun__imageE,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod(A,B),F3: fun(C,A),G3: fun(D,B),R2: set(product_prod(C,D))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),C3),image2(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F3,G3),R2)))
     => ~ ! [X3: C,Y3: D] :
            ( ( C3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,X3)),aa(D,B,G3,Y3)) )
           => ~ pp(aa(set(product_prod(C,D)),bool,aa(product_prod(C,D),fun(set(product_prod(C,D)),bool),member(product_prod(C,D)),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),X3),Y3)),R2)) ) ) ).

% prod_fun_imageE
tff(fact_5889_map__prod__def,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: fun(A,C),G3: fun(B,D)] : product_map_prod(A,C,B,D,F3,G3) = aa(fun(A,fun(B,product_prod(C,D))),fun(product_prod(A,B),product_prod(C,D)),product_case_prod(A,B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_abq(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),F3),G3)) ).

% map_prod_def
tff(fact_5890_map__prod__surj,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: fun(A,B),G3: fun(C,D)] :
      ( ( image2(A,B,F3,top_top(set(A))) = top_top(set(B)) )
     => ( ( image2(C,D,G3,top_top(set(C))) = top_top(set(D)) )
       => ( image2(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F3,G3),top_top(set(product_prod(A,C)))) = top_top(set(product_prod(B,D))) ) ) ) ).

% map_prod_surj
tff(fact_5891_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A12: nat,A23: nat,A32: A,P2: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))))] :
      ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A12),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),A23),A32))))
     => ( ! [F2: fun(nat,fun(A,A)),A5: nat,B4: nat,Acc: A] :
            ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),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)),A5),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B4),Acc))))
           => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B4),A5))
               => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P2,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),one_one(nat))),B4),aa(A,A,aa(nat,fun(A,A),F2,A5),Acc))) )
             => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P2,F2),A5),B4),Acc)) ) )
       => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P2,A0),A12),A23),A32)) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_5892_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa2: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa2,Xb,Xc) = Y )
     => ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc))))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa2))
               => ( Y = Xc ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa2))
               => ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa2),Xc)) ) ) )
           => ~ accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_5893_tendsto__iff__uniformity,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo7287701948861334536_space(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [E6: fun(product_prod(B,B),bool)] :
              ( eventually(product_prod(B,B),E6,topolo7806501430040627800ormity(B))
             => eventually(A,aa(fun(product_prod(B,B),bool),fun(A,bool),aa(B,fun(fun(product_prod(B,B),bool),fun(A,bool)),aTP_Lamp_abr(fun(A,B),fun(B,fun(fun(product_prod(B,B),bool),fun(A,bool))),F3),L),E6),F4) ) ) ) ).

% tendsto_iff_uniformity
tff(fact_5894_independentD,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A),T2: set(A),U: fun(A,real),V3: A] :
          ( ~ real_V358717886546972837endent(A,S2)
         => ( pp(aa(set(A),bool,finite_finite2(A),T2))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S2))
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_abs(fun(A,real),fun(A,A),U)),T2) = zero_zero(A) )
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V3),T2))
                 => ( aa(A,real,U,V3) = zero_zero(real) ) ) ) ) ) ) ) ).

% independentD
tff(fact_5895_dependent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B6: set(A)] :
          ( real_V358717886546972837endent(A,B6)
        <=> ? [X7: fun(A,real)] :
              ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_abt(fun(A,real),fun(A,bool),X7))))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_abt(fun(A,real),fun(A,bool),X7))),B6))
              & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_abs(fun(A,real),fun(A,A),X7)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_abt(fun(A,real),fun(A,bool),X7))) = zero_zero(A) )
              & ? [X4: A] : aa(A,real,X7,X4) != zero_zero(real) ) ) ) ).

% dependent_alt
tff(fact_5896_dependent__single,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] :
          ( real_V358717886546972837endent(A,aa(set(A),set(A),insert(A,X),bot_bot(set(A))))
        <=> ( X = zero_zero(A) ) ) ) ).

% dependent_single
tff(fact_5897_uniformity__transE,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),bool)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => ~ ! [D8: fun(product_prod(A,A),bool)] :
                ( eventually(product_prod(A,A),D8,topolo7806501430040627800ormity(A))
               => ~ ! [X5: A,Y4: A] :
                      ( pp(aa(product_prod(A,A),bool,D8,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y4)))
                     => ! [Z4: A] :
                          ( pp(aa(product_prod(A,A),bool,D8,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)))
                         => pp(aa(product_prod(A,A),bool,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Z4))) ) ) ) ) ) ).

% uniformity_transE
tff(fact_5898_uniformity__trans,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),bool)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => ? [D8: fun(product_prod(A,A),bool)] :
              ( eventually(product_prod(A,A),D8,topolo7806501430040627800ormity(A))
              & ! [X5: A,Y4: A,Z4: A] :
                  ( pp(aa(product_prod(A,A),bool,D8,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y4)))
                 => ( pp(aa(product_prod(A,A),bool,D8,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)))
                   => pp(aa(product_prod(A,A),bool,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Z4))) ) ) ) ) ) ).

% uniformity_trans
tff(fact_5899_uniformity__refl,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),bool),X: A] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => pp(aa(product_prod(A,A),bool,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X))) ) ) ).

% uniformity_refl
tff(fact_5900_dependent__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),A6))
         => real_V358717886546972837endent(A,A6) ) ) ).

% dependent_zero
tff(fact_5901_uniformity__sym,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),bool)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => eventually(product_prod(A,A),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_abu(fun(product_prod(A,A),bool),fun(A,fun(A,bool)),E5)),topolo7806501430040627800ormity(A)) ) ) ).

% uniformity_sym
tff(fact_5902_Cauchy__uniform__iff,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [P6: fun(product_prod(A,A),bool)] :
              ( eventually(product_prod(A,A),P6,topolo7806501430040627800ormity(A))
             => ? [N7: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N5))
                 => ! [M6: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),M6))
                     => pp(aa(product_prod(A,A),bool,P6,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,X6,N5)),aa(nat,A,X6,M6)))) ) ) ) ) ) ).

% Cauchy_uniform_iff
tff(fact_5903_dependent__finite,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S3))
         => ( real_V358717886546972837endent(A,S3)
          <=> ? [U5: fun(A,real)] :
                ( ? [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
                    & ( aa(A,real,U5,X4) != zero_zero(real) ) )
                & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_abs(fun(A,real),fun(A,A),U5)),S3) = zero_zero(A) ) ) ) ) ) ).

% dependent_finite
tff(fact_5904_independent__if__scalars__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ! [F2: fun(A,real),X3: A] :
                ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_abs(fun(A,real),fun(A,A),F2)),A6) = zero_zero(A) )
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
                 => ( aa(A,real,F2,X3) = zero_zero(real) ) ) )
           => ~ real_V358717886546972837endent(A,A6) ) ) ) ).

% independent_if_scalars_zero
tff(fact_5905_eventually__uniformity__metric,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [P2: fun(product_prod(A,A),bool)] :
          ( eventually(product_prod(A,A),P2,topolo7806501430040627800ormity(A))
        <=> ? [E4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E4))
              & ! [X4: A,Y5: A] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X4,Y5)),E4))
                 => pp(aa(product_prod(A,A),bool,P2,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5))) ) ) ) ) ).

% eventually_uniformity_metric
tff(fact_5906_independent__explicit__finite__subsets,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A6: set(A)] :
          ( ~ real_V358717886546972837endent(A,A6)
        <=> ! [S6: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S6),A6))
             => ( pp(aa(set(A),bool,finite_finite2(A),S6))
               => ! [U5: fun(A,real)] :
                    ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_abs(fun(A,real),fun(A,A),U5)),S6) = zero_zero(A) )
                   => ! [X4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S6))
                       => ( aa(A,real,U5,X4) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_finite_subsets
tff(fact_5907_independent__explicit__module,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] :
          ( ~ real_V358717886546972837endent(A,S2)
        <=> ! [T8: set(A),U5: fun(A,real),V4: A] :
              ( pp(aa(set(A),bool,finite_finite2(A),T8))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T8),S2))
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_abs(fun(A,real),fun(A,A),U5)),T8) = zero_zero(A) )
                 => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V4),T8))
                   => ( aa(A,real,U5,V4) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_module
tff(fact_5908_dependent__explicit,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] :
          ( real_V358717886546972837endent(A,S2)
        <=> ? [T8: set(A)] :
              ( pp(aa(set(A),bool,finite_finite2(A),T8))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T8),S2))
              & ? [U5: fun(A,real)] :
                  ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_abs(fun(A,real),fun(A,A),U5)),T8) = zero_zero(A) )
                  & ? [X4: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),T8))
                      & ( aa(A,real,U5,X4) != zero_zero(real) ) ) ) ) ) ) ).

% dependent_explicit
tff(fact_5909_independentD__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B6: set(A),X6: fun(A,real),X: A] :
          ( ~ real_V358717886546972837endent(A,B6)
         => ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_abt(fun(A,real),fun(A,bool),X6))))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_abt(fun(A,real),fun(A,bool),X6))),B6))
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_abs(fun(A,real),fun(A,A),X6)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_abt(fun(A,real),fun(A,bool),X6))) = zero_zero(A) )
               => ( aa(A,real,X6,X) = zero_zero(real) ) ) ) ) ) ) ).

% independentD_alt
tff(fact_5910_independent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B6: set(A)] :
          ( ~ real_V358717886546972837endent(A,B6)
        <=> ! [X7: fun(A,real)] :
              ( pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_abt(fun(A,real),fun(A,bool),X7))))
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_abt(fun(A,real),fun(A,bool),X7))),B6))
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_abs(fun(A,real),fun(A,A),X7)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_abt(fun(A,real),fun(A,bool),X7))) = zero_zero(A) )
                 => ! [X4: A] : aa(A,real,X7,X4) = zero_zero(real) ) ) ) ) ) ).

% independent_alt
tff(fact_5911_totally__bounded__def,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S3: set(A)] :
          ( topolo6688025880775521714ounded(A,S3)
        <=> ! [E6: fun(product_prod(A,A),bool)] :
              ( eventually(product_prod(A,A),E6,topolo7806501430040627800ormity(A))
             => ? [X7: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite2(A),X7))
                  & ! [X4: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
                     => ? [Xa3: A] :
                          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),X7))
                          & pp(aa(product_prod(A,A),bool,E6,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X4))) ) ) ) ) ) ) ).

% totally_bounded_def
tff(fact_5912_uniformity__trans_H,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),bool)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => eventually(product_prod(product_prod(A,A),product_prod(A,A)),aa(fun(product_prod(A,A),fun(product_prod(A,A),bool)),fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),product_case_prod(product_prod(A,A),product_prod(A,A),bool),aa(fun(A,fun(A,fun(product_prod(A,A),bool))),fun(product_prod(A,A),fun(product_prod(A,A),bool)),product_case_prod(A,A,fun(product_prod(A,A),bool)),aTP_Lamp_abw(fun(product_prod(A,A),bool),fun(A,fun(A,fun(product_prod(A,A),bool))),E5))),prod_filter(product_prod(A,A),product_prod(A,A),topolo7806501430040627800ormity(A),topolo7806501430040627800ormity(A))) ) ) ).

% uniformity_trans'
tff(fact_5913_uniformly__continuous__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [S2: set(A),F3: fun(A,B),E5: fun(product_prod(B,B),bool)] :
          ( topolo6026614971017936543ous_on(A,B,S2,F3)
         => ( eventually(product_prod(B,B),E5,topolo7806501430040627800ormity(B))
           => eventually(product_prod(A,A),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(product_prod(B,B),bool),fun(A,fun(A,bool)),aa(fun(A,B),fun(fun(product_prod(B,B),bool),fun(A,fun(A,bool))),aTP_Lamp_abx(set(A),fun(fun(A,B),fun(fun(product_prod(B,B),bool),fun(A,fun(A,bool)))),S2),F3),E5)),topolo7806501430040627800ormity(A)) ) ) ) ).

% uniformly_continuous_onD
tff(fact_5914_eventually__prod__filter,axiom,
    ! [A: $tType,B: $tType,P2: fun(product_prod(A,B),bool),F4: filter(A),G7: filter(B)] :
      ( eventually(product_prod(A,B),P2,prod_filter(A,B,F4,G7))
    <=> ? [Pf: fun(A,bool),Pg: fun(B,bool)] :
          ( eventually(A,Pf,F4)
          & eventually(B,Pg,G7)
          & ! [X4: A,Y5: B] :
              ( pp(aa(A,bool,Pf,X4))
             => ( pp(aa(B,bool,Pg,Y5))
               => pp(aa(product_prod(A,B),bool,P2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y5))) ) ) ) ) ).

% eventually_prod_filter
tff(fact_5915_eventually__prod__same,axiom,
    ! [A: $tType,P2: fun(product_prod(A,A),bool),F4: filter(A)] :
      ( eventually(product_prod(A,A),P2,prod_filter(A,A,F4,F4))
    <=> ? [Q6: fun(A,bool)] :
          ( eventually(A,Q6,F4)
          & ! [X4: A,Y5: A] :
              ( pp(aa(A,bool,Q6,X4))
             => ( pp(aa(A,bool,Q6,Y5))
               => pp(aa(product_prod(A,A),bool,P2,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5))) ) ) ) ) ).

% eventually_prod_same
tff(fact_5916_nhds__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [A3: A,B2: B] : topolo7230453075368039082e_nhds(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) = prod_filter(A,B,topolo7230453075368039082e_nhds(A,A3),topolo7230453075368039082e_nhds(B,B2)) ) ).

% nhds_prod
tff(fact_5917_filterlim__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,B),G7: filter(B),F4: filter(A),G3: fun(A,C),H5: filter(C)] :
      ( filterlim(A,B,F3,G7,F4)
     => ( filterlim(A,C,G3,H5,F4)
       => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aby(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3),prod_filter(B,C,G7,H5),F4) ) ) ).

% filterlim_Pair
tff(fact_5918_tendsto__mult__Pair,axiom,
    ! [A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [A3: A,B2: A] : filterlim(product_prod(A,A),A,aTP_Lamp_abz(product_prod(A,A),A),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),prod_filter(A,A,topolo7230453075368039082e_nhds(A,A3),topolo7230453075368039082e_nhds(A,B2))) ) ).

% tendsto_mult_Pair
tff(fact_5919_prod__filter__assoc,axiom,
    ! [A: $tType,B: $tType,C: $tType,F4: filter(A),G7: filter(B),H5: filter(C)] : prod_filter(product_prod(A,B),C,prod_filter(A,B,F4,G7),H5) = filtermap(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C),aa(fun(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),fun(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),product_case_prod(A,product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_acb(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)))),prod_filter(A,product_prod(B,C),F4,prod_filter(B,C,G7,H5))) ).

% prod_filter_assoc
tff(fact_5920_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),N: nat] :
          ( pp(bit_se6407376104438227557le_bit(A,Tyrep,N))
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_5921_acyclic__insert,axiom,
    ! [A: $tType,Y: A,X: A,R: set(product_prod(A,A))] :
      ( transitive_acyclic(A,aa(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),Y),X)),R))
    <=> ( transitive_acyclic(A,R)
        & ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R))) ) ) ).

% acyclic_insert
tff(fact_5922_filtermap__Pair,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),G3: fun(C,B),F4: filter(C)] : pp(aa(filter(product_prod(A,B)),bool,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),bool),ord_less_eq(filter(product_prod(A,B))),filtermap(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_acc(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F3),G3),F4)),prod_filter(A,B,filtermap(C,A,F3,F4),filtermap(C,B,G3,F4)))) ).

% filtermap_Pair
tff(fact_5923_possible__bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ty: itself(A)] : pp(bit_se6407376104438227557le_bit(A,Ty,zero_zero(nat))) ) ).

% possible_bit_0
tff(fact_5924_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I2: nat,J2: nat] :
          ( pp(bit_se6407376104438227557le_bit(A,Tyrep,I2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I2))
           => pp(bit_se6407376104438227557le_bit(A,Tyrep,J2)) ) ) ) ).

% possible_bit_less_imp
tff(fact_5925_acyclicI__order,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [R: set(product_prod(B,B)),F3: fun(B,A)] :
          ( ! [A5: B,B4: B] :
              ( pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A5),B4)),R))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,B4)),aa(B,A,F3,A5))) )
         => transitive_acyclic(B,R) ) ) ).

% acyclicI_order
tff(fact_5926_filtermap__nhds__times,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,A3: A] :
          ( ( C3 != zero_zero(A) )
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C3),topolo7230453075368039082e_nhds(A,A3)) = topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)) ) ) ) ).

% filtermap_nhds_times
tff(fact_5927_acyclic__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( transitive_acyclic(A,R)
    <=> ! [X4: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),transitive_trancl(A,R))) ) ).

% acyclic_def
tff(fact_5928_acyclicI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),transitive_trancl(A,R)))
     => transitive_acyclic(A,R) ) ).

% acyclicI
tff(fact_5929_eventually__prod__sequentially,axiom,
    ! [P2: fun(product_prod(nat,nat),bool)] :
      ( eventually(product_prod(nat,nat),P2,prod_filter(nat,nat,at_top(nat),at_top(nat)))
    <=> ? [N7: nat] :
        ! [M6: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),M6))
         => ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N5))
             => pp(aa(product_prod(nat,nat),bool,P2,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N5),M6))) ) ) ) ).

% eventually_prod_sequentially
tff(fact_5930_at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A] : topolo174197925503356063within(A,A3,top_top(set(A))) = filtermap(A,A,aTP_Lamp_acd(A,fun(A,A),A3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% at_to_0
tff(fact_5931_filtermap__times__pos__at__right,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [C3: A,P: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C3),topolo174197925503356063within(A,P,aa(A,set(A),set_ord_greaterThan(A),P))) = topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),P),aa(A,set(A),set_ord_greaterThan(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),P))) ) ) ) ).

% filtermap_times_pos_at_right
tff(fact_5932_at__to__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ( topolo174197925503356063within(A,zero_zero(A),top_top(set(A))) = filtermap(A,A,inverse_inverse(A),at_infinity(A)) ) ) ).

% at_to_infinity
tff(fact_5933_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,N: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,M2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N),bit_se6407376104438227557le_bit(A,type2(A),N)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ).

% drop_bit_exp_eq
tff(fact_5934_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2)))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% bit_minus_2_iff
tff(fact_5935_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))
            & ( N != zero_zero(nat) )
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ) ).

% bit_double_iff
tff(fact_5936_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_5937_of__nat__CHAR,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),semiri4206861660011772517g_char(A,type2(A))) = zero_zero(A) ) ) ).

% of_nat_CHAR
tff(fact_5938_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C3: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C3) = zero_zero(A) )
         => ( ! [X3: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X3) = zero_zero(A) )
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),C3),X3)) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C3 ) ) ) ) ).

% CHAR_eqI
tff(fact_5939_of__nat__eq__0__iff__char__dvd,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),semiri4206861660011772517g_char(A,type2(A))),N)) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_5940_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A))))
      <=> ? [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
            & ( aa(nat,A,semiring_1_of_nat(A),N5) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_5941_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C3))
         => ( ( aa(nat,A,semiring_1_of_nat(A),C3) = zero_zero(A) )
           => ( ! [X3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X3))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),C3))
                   => ( aa(nat,A,semiring_1_of_nat(A),X3) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C3 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_5942_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
           => ( aa(nat,A,semiring_1_of_nat(A),N5) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_5943_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M2: nat,A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4730199178511100633sh_bit(A,M2),A3)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
            & pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).

% bit_push_bit_iff
tff(fact_5944_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
        <=> ~ pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ).

% fold_possible_bit
tff(fact_5945_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M2: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M2))),N))
        <=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ) ) ).

% bit_minus_exp_iff
tff(fact_5946_prod__filter__principal__singleton2,axiom,
    ! [B: $tType,A: $tType,F4: filter(A),X: B] : prod_filter(A,B,F4,principal(B,aa(set(B),set(B),insert(B,X),bot_bot(set(B))))) = filtermap(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_mv(B,fun(A,product_prod(A,B))),X),F4) ).

% prod_filter_principal_singleton2
tff(fact_5947_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = aa(option(num),A,aa(fun(num,A),fun(option(num),A),aa(A,fun(fun(num,A),fun(option(num),A)),case_option(A,num),zero_zero(A)),numeral_numeral(A)),bit_un2480387367778600638or_num(M2,N)) ) ).

% numeral_xor_num
tff(fact_5948_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = aa(option(num),A,aa(fun(num,A),fun(option(num),A),aa(A,fun(fun(num,A),fun(option(num),A)),case_option(A,num),zero_zero(A)),numeral_numeral(A)),bit_un7362597486090784418nd_num(M2,N)) ) ).

% numeral_and_num
tff(fact_5949_and__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: num,N: num] :
          ( ( bit_un7362597486090784418nd_num(M2,N) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% and_num_eq_None_iff
tff(fact_5950_xor__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M2: num,N: num] :
          ( ( bit_un2480387367778600638or_num(M2,N) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M2)),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% xor_num_eq_None_iff
tff(fact_5951_prod__filter__principal__singleton,axiom,
    ! [A: $tType,B: $tType,X: A,F4: filter(B)] : prod_filter(A,B,principal(A,aa(set(A),set(A),insert(A,X),bot_bot(set(A)))),F4) = filtermap(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),F4) ).

% prod_filter_principal_singleton
tff(fact_5952_last__list__update,axiom,
    ! [A: $tType,Xs: list(A),K2: nat,X: A] :
      ( ( Xs != nil(A) )
     => ( ( ( K2 = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
         => ( last(A,list_update(A,Xs,K2,X)) = X ) )
        & ( ( K2 != aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
         => ( last(A,list_update(A,Xs,K2,X)) = last(A,Xs) ) ) ) ) ).

% last_list_update
tff(fact_5953_init__seg__of__def,axiom,
    ! [A: $tType] : init_seg_of(A) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),bool),aTP_Lamp_ace(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)))) ).

% init_seg_of_def
tff(fact_5954_last__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( last(A,replicate(A,N,X)) = X ) ) ).

% last_replicate
tff(fact_5955_last__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( last(A,drop(A,N,Xs)) = last(A,Xs) ) ) ).

% last_drop
tff(fact_5956_last__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
         => ( last(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),last(A,Xs)),last(B,Ys)) ) ) ) ) ).

% last_zip
tff(fact_5957_last__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( last(A,Xs) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))) ) ) ).

% last_conv_nth
tff(fact_5958_sqr_Osimps_I3_J,axiom,
    ! [N: num] : sqr(aa(num,num,bit1,N)) = aa(num,num,bit1,bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),sqr(N)),N))) ).

% sqr.simps(3)
tff(fact_5959_arg__min__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F3: fun(A,B),P2: fun(A,bool),A3: A] :
          ( inj_on(A,B,F3,aa(fun(A,bool),set(A),collect(A),P2))
         => ( pp(aa(A,bool,P2,A3))
           => ( ! [Y3: A] :
                  ( pp(aa(A,bool,P2,Y3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,A3)),aa(A,B,F3,Y3))) )
             => ( lattices_ord_arg_min(A,B,F3,P2) = A3 ) ) ) ) ) ).

% arg_min_inj_eq
tff(fact_5960_arg__min__nat__le,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,M2: fun(A,nat)] :
      ( pp(aa(A,bool,P2,X))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M2,lattices_ord_arg_min(A,nat,M2,P2))),aa(A,nat,M2,X))) ) ).

% arg_min_nat_le
tff(fact_5961_arg__min__nat__lemma,axiom,
    ! [A: $tType,P2: fun(A,bool),K2: A,M2: fun(A,nat)] :
      ( pp(aa(A,bool,P2,K2))
     => ( pp(aa(A,bool,P2,lattices_ord_arg_min(A,nat,M2,P2)))
        & ! [Y4: A] :
            ( pp(aa(A,bool,P2,Y4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M2,lattices_ord_arg_min(A,nat,M2,P2))),aa(A,nat,M2,Y4))) ) ) ) ).

% arg_min_nat_lemma
tff(fact_5962_sqr_Osimps_I2_J,axiom,
    ! [N: num] : sqr(bit0(N)) = bit0(bit0(sqr(N))) ).

% sqr.simps(2)
tff(fact_5963_sqr_Osimps_I1_J,axiom,
    sqr(one2) = one2 ).

% sqr.simps(1)
tff(fact_5964_sqr__conv__mult,axiom,
    ! [X: num] : sqr(X) = aa(num,num,aa(num,fun(num,num),times_times(num),X),X) ).

% sqr_conv_mult
tff(fact_5965_arg__min__equality,axiom,
    ! [A: $tType,C: $tType] :
      ( order(A)
     => ! [P2: fun(C,bool),K2: C,F3: fun(C,A)] :
          ( pp(aa(C,bool,P2,K2))
         => ( ! [X3: C] :
                ( pp(aa(C,bool,P2,X3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F3,K2)),aa(C,A,F3,X3))) )
           => ( aa(C,A,F3,lattices_ord_arg_min(C,A,F3,P2)) = aa(C,A,F3,K2) ) ) ) ) ).

% arg_min_equality
tff(fact_5966_numeral__sqr,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K2: num] : aa(num,A,numeral_numeral(A),sqr(K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K2)),aa(num,A,numeral_numeral(A),K2)) ) ).

% numeral_sqr
tff(fact_5967_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] : pow(X,aa(num,num,bit1,Y)) = aa(num,num,aa(num,fun(num,num),times_times(num),sqr(pow(X,Y))),X) ).

% pow.simps(3)
tff(fact_5968_refl__on__singleton,axiom,
    ! [A: $tType,X: A] : refl_on(A,aa(set(A),set(A),insert(A,X),bot_bot(set(A))),aa(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),X),X)),bot_bot(set(product_prod(A,A))))) ).

% refl_on_singleton
tff(fact_5969_refl__onD,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,A6,R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R)) ) ) ).

% refl_onD
tff(fact_5970_refl__onD1,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X: A,Y: A] :
      ( refl_on(A,A6,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6)) ) ) ).

% refl_onD1
tff(fact_5971_refl__onD2,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X: A,Y: A] :
      ( refl_on(A,A6,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A6)) ) ) ).

% refl_onD2
tff(fact_5972_refl__on__domain,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
      ( refl_on(A,A6,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A6)) ) ) ) ).

% refl_on_domain
tff(fact_5973_pow_Osimps_I1_J,axiom,
    ! [X: num] : pow(X,one2) = X ).

% pow.simps(1)
tff(fact_5974_refl__on__def_H,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
      ( refl_on(A,A6,R)
    <=> ( ! [X4: product_prod(A,A)] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),X4),R))
           => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_acf(set(A),fun(A,fun(A,bool)),A6)),X4)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R)) ) ) ) ).

% refl_on_def'
tff(fact_5975_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] : pow(X,bit0(Y)) = sqr(pow(X,Y)) ).

% pow.simps(2)
tff(fact_5976_wo__rel_Ocases__Total3,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))),field2(A,R)))
       => ( ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A))))
              | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
         => ( ( ( A3 = B2 )
             => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_5977_Ex__inj__on__UNION__Sigma,axiom,
    ! [A: $tType,B: $tType,A6: fun(B,set(A)),I5: set(B)] :
    ? [F2: fun(A,product_prod(B,A))] :
      ( inj_on(A,product_prod(B,A),F2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(B,set(A),A6,I5)))
      & pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),image2(A,product_prod(B,A),F2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(B,set(A),A6,I5)))),product_Sigma(B,A,I5,A6))) ) ).

% Ex_inj_on_UNION_Sigma
tff(fact_5978_mem__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A6: set(A),B6: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B6)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
        & pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),aa(A,set(B),B6,A3))) ) ) ).

% mem_Sigma_iff
tff(fact_5979_SigmaI,axiom,
    ! [B: $tType,A: $tType,A3: A,A6: set(A),B2: B,B6: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),aa(A,set(B),B6,A3)))
       => pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B6))) ) ) ).

% SigmaI
tff(fact_5980_Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,bool),Q: fun(B,bool)] : aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_acg(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),P2),Q))) = product_Sigma(A,B,aa(fun(A,bool),set(A),collect(A),P2),aTP_Lamp_ach(fun(B,bool),fun(A,set(B)),Q)) ).

% Collect_case_prod
tff(fact_5981_Sigma__empty1,axiom,
    ! [B: $tType,A: $tType,B6: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B6) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty1
tff(fact_5982_Compl__Times__UNIV1,axiom,
    ! [A: $tType,B: $tType,A6: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_aci(set(B),fun(A,set(B)),A6))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_acj(set(B),fun(A,set(B)),A6)) ).

% Compl_Times_UNIV1
tff(fact_5983_Compl__Times__UNIV2,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_ack(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A6),aTP_Lamp_ack(A,set(B))) ).

% Compl_Times_UNIV2
tff(fact_5984_Sigma__empty2,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] : product_Sigma(A,B,A6,aTP_Lamp_acl(A,set(B))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty2
tff(fact_5985_Times__empty,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( ( product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6)) = bot_bot(set(product_prod(A,B))) )
    <=> ( ( A6 = bot_bot(set(A)) )
        | ( B6 = bot_bot(set(B)) ) ) ) ).

% Times_empty
tff(fact_5986_UNIV__Times__UNIV,axiom,
    ! [B: $tType,A: $tType] : product_Sigma(A,B,top_top(set(A)),aTP_Lamp_ack(A,set(B))) = top_top(set(product_prod(A,B))) ).

% UNIV_Times_UNIV
tff(fact_5987_fst__image__times,axiom,
    ! [B: $tType,A: $tType,B6: set(B),A6: set(A)] :
      ( ( ( B6 = bot_bot(set(B)) )
       => ( image2(product_prod(A,B),A,product_fst(A,B),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6))) = bot_bot(set(A)) ) )
      & ( ( B6 != bot_bot(set(B)) )
       => ( image2(product_prod(A,B),A,product_fst(A,B),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6))) = A6 ) ) ) ).

% fst_image_times
tff(fact_5988_snd__image__times,axiom,
    ! [B: $tType,A: $tType,A6: set(B),B6: set(A)] :
      ( ( ( A6 = bot_bot(set(B)) )
       => ( image2(product_prod(B,A),A,product_snd(B,A),product_Sigma(B,A,A6,aTP_Lamp_acm(set(A),fun(B,set(A)),B6))) = bot_bot(set(A)) ) )
      & ( ( A6 != bot_bot(set(B)) )
       => ( image2(product_prod(B,A),A,product_snd(B,A),product_Sigma(B,A,A6,aTP_Lamp_acm(set(A),fun(B,set(A)),B6))) = B6 ) ) ) ).

% snd_image_times
tff(fact_5989_insert__Times__insert,axiom,
    ! [A: $tType,B: $tType,A3: A,A6: set(A),B2: B,B6: set(B)] : product_Sigma(A,B,aa(set(A),set(A),insert(A,A3),A6),aa(set(B),fun(A,set(B)),aTP_Lamp_acn(B,fun(set(B),fun(A,set(B))),B2),B6)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A6,aa(set(B),fun(A,set(B)),aTP_Lamp_acn(B,fun(set(B),fun(A,set(B))),B2),B6))),product_Sigma(A,B,aa(set(A),set(A),insert(A,A3),A6),aTP_Lamp_aci(set(B),fun(A,set(B)),B6)))) ).

% insert_Times_insert
tff(fact_5990_inj__on__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C),A6: set(A)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F3),product_Sigma(A,B,A6,aTP_Lamp_ack(A,set(B))))
    <=> inj_on(A,C,F3,A6) ) ).

% inj_on_apfst
tff(fact_5991_inj__on__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C),A6: set(B)] :
      ( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_aci(set(B),fun(A,set(B)),A6)))
    <=> inj_on(B,C,F3,A6) ) ).

% inj_on_apsnd
tff(fact_5992_Collect__case__prod__Sigma,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,bool),Q: fun(A,fun(B,bool))] : aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_aco(fun(A,bool),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P2),Q))) = product_Sigma(A,B,aa(fun(A,bool),set(A),collect(A),P2),aTP_Lamp_acp(fun(A,fun(B,bool)),fun(A,set(B)),Q)) ).

% Collect_case_prod_Sigma
tff(fact_5993_Sigma__mono,axiom,
    ! [B: $tType,A: $tType,A6: set(A),C5: set(A),B6: fun(A,set(B)),D5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C5))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B6,X3)),aa(A,set(B),D5,X3))) )
       => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A6,B6)),product_Sigma(A,B,C5,D5))) ) ) ).

% Sigma_mono
tff(fact_5994_Sigma__Int__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),I5),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I5,C5)),product_Sigma(A,B,J4,C5)) ).

% Sigma_Int_distrib1
tff(fact_5995_Sigma__Diff__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I5,C5)),product_Sigma(A,B,J4,C5)) ).

% Sigma_Diff_distrib1
tff(fact_5996_Sigma__Un__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I5),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I5,C5)),product_Sigma(A,B,J4,C5)) ).

% Sigma_Un_distrib1
tff(fact_5997_Times__eq__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C5: set(A),A6: set(B),B6: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),C5))
     => ( ( product_Sigma(B,A,A6,aTP_Lamp_acm(set(A),fun(B,set(A)),C5)) = product_Sigma(B,A,B6,aTP_Lamp_acm(set(A),fun(B,set(A)),C5)) )
      <=> ( A6 = B6 ) ) ) ).

% Times_eq_cancel2
tff(fact_5998_Sigma__cong,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(A),C5: fun(A,set(B)),D5: fun(A,set(B))] :
      ( ( A6 = B6 )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B6))
           => ( aa(A,set(B),C5,X3) = aa(A,set(B),D5,X3) ) )
       => ( product_Sigma(A,B,A6,C5) = product_Sigma(A,B,B6,D5) ) ) ) ).

% Sigma_cong
tff(fact_5999_times__eq__iff,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B),C5: set(A),D5: set(B)] :
      ( ( product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6)) = product_Sigma(A,B,C5,aTP_Lamp_aci(set(B),fun(A,set(B)),D5)) )
    <=> ( ( ( A6 = C5 )
          & ( B6 = D5 ) )
        | ( ( ( A6 = bot_bot(set(A)) )
            | ( B6 = bot_bot(set(B)) ) )
          & ( ( C5 = bot_bot(set(A)) )
            | ( D5 = bot_bot(set(B)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_6000_Product__Type_Oproduct__def,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] : product_product(A,B,A6,B6) = product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6)) ).

% Product_Type.product_def
tff(fact_6001_member__product,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A6: set(A),B6: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X),product_product(A,B,A6,B6)))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6)))) ) ).

% member_product
tff(fact_6002_wo__rel_Omax2__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
         => ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = B2 ) )
        & ( ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
         => ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = A3 ) ) ) ) ).

% wo_rel.max2_def
tff(fact_6003_wo__rel_OTOTALS,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ! [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),field2(A,R)))
         => ! [Xa: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),field2(A,R)))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa)),R))
                | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X5)),R)) ) ) ) ) ).

% wo_rel.TOTALS
tff(fact_6004_SigmaE2,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A6: set(A),B6: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B6)))
     => ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
         => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),aa(A,set(B),B6,A3))) ) ) ).

% SigmaE2
tff(fact_6005_SigmaD2,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A6: set(A),B6: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B6)))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),aa(A,set(B),B6,A3))) ) ).

% SigmaD2
tff(fact_6006_SigmaD1,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,A6: set(A),B6: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B6)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6)) ) ).

% SigmaD1
tff(fact_6007_SigmaE,axiom,
    ! [A: $tType,B: $tType,C3: product_prod(A,B),A6: set(A),B6: fun(A,set(B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),C3),product_Sigma(A,B,A6,B6)))
     => ~ ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
           => ! [Y3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y3),aa(A,set(B),B6,X3)))
               => ( C3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3) ) ) ) ) ).

% SigmaE
tff(fact_6008_wo__rel_Owell__order__induct,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P2: fun(A,bool),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( ! [X3: A] :
            ( ! [Y4: A] :
                ( ( ( Y4 != X3 )
                  & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R)) )
               => pp(aa(A,bool,P2,Y4)) )
           => pp(aa(A,bool,P2,X3)) )
       => pp(aa(A,bool,P2,A3)) ) ) ).

% wo_rel.well_order_induct
tff(fact_6009_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C5: set(A),A6: set(B),B6: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),C5))
     => ( pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A6,aTP_Lamp_acm(set(A),fun(B,set(A)),C5))),product_Sigma(B,A,B6,aTP_Lamp_acm(set(A),fun(B,set(A)),C5))))
      <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B6)) ) ) ).

% Times_subset_cancel2
tff(fact_6010_mem__Times__iff,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),A6: set(A),B6: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6))))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(product_prod(A,B),A,product_fst(A,B),X)),A6))
        & pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(product_prod(A,B),B,product_snd(A,B),X)),B6)) ) ) ).

% mem_Times_iff
tff(fact_6011_Sigma__empty__iff,axiom,
    ! [A: $tType,B: $tType,I5: set(A),X6: fun(A,set(B))] :
      ( ( product_Sigma(A,B,I5,X6) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),I5))
         => ( aa(A,set(B),X6,X4) = bot_bot(set(B)) ) ) ) ).

% Sigma_empty_iff
tff(fact_6012_Times__Int__distrib1,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6),aTP_Lamp_aci(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B6,aTP_Lamp_aci(set(B),fun(A,set(B)),C5))) ).

% Times_Int_distrib1
tff(fact_6013_Sigma__Int__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B)),B6: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_acq(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A6),B6)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I5,A6)),product_Sigma(A,B,I5,B6)) ).

% Sigma_Int_distrib2
tff(fact_6014_Times__Int__Times,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B),C5: set(A),D5: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6))),product_Sigma(A,B,C5,aTP_Lamp_aci(set(B),fun(A,set(B)),D5))) = product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),C5),aa(set(B),fun(A,set(B)),aTP_Lamp_acr(set(B),fun(set(B),fun(A,set(B))),B6),D5)) ).

% Times_Int_Times
tff(fact_6015_infinite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ~ pp(aa(set(B),bool,finite_finite2(B),B6))
       => ~ pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6)))) ) ) ).

% infinite_cartesian_product
tff(fact_6016_wo__rel_Omax2__equals1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R)))
         => ( ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = A3 )
          <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R)) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_6017_wo__rel_Omax2__equals2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R)))
         => ( ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = B2 )
          <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_6018_wo__rel_Omax2__greater,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R)))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R,A3,B2))),R))
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R,A3,B2))),R)) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_6019_Sigma__Un__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B)),B6: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_acs(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A6),B6)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I5,A6)),product_Sigma(A,B,I5,B6)) ).

% Sigma_Un_distrib2
tff(fact_6020_Times__Un__distrib1,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B6),aTP_Lamp_aci(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B6,aTP_Lamp_aci(set(B),fun(A,set(B)),C5))) ).

% Times_Un_distrib1
tff(fact_6021_Sigma__Union,axiom,
    ! [B: $tType,A: $tType,X6: set(set(A)),B6: fun(A,set(B))] : product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X6),B6) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image2(set(A),set(product_prod(A,B)),aTP_Lamp_act(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),B6),X6)) ).

% Sigma_Union
tff(fact_6022_Sigma__Diff__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B)),B6: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_acu(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A6),B6)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I5,A6)),product_Sigma(A,B,I5,B6)) ).

% Sigma_Diff_distrib2
tff(fact_6023_Times__Diff__distrib1,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B6),aTP_Lamp_aci(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B6,aTP_Lamp_aci(set(B),fun(A,set(B)),C5))) ).

% Times_Diff_distrib1
tff(fact_6024_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A6: set(A),C5: set(B),B6: set(A),D5: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B6,aTP_Lamp_aci(set(B),fun(A,set(B)),D5))))
    <=> ( ( A6 = bot_bot(set(A)) )
        | ( C5 = bot_bot(set(B)) )
        | ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C5),D5)) ) ) ) ).

% times_subset_iff
tff(fact_6025_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A6,aTP_Lamp_acv(set(A),fun(A,set(A)),A6))))
       => ( ( A3 = B2 )
          | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6)) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_6026_wfI,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A6: set(A),B6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A6,aTP_Lamp_acv(set(A),fun(A,set(A)),B6))))
     => ( ! [X3: A,P5: fun(A,bool)] :
            ( ! [Xa: A] :
                ( ! [Y3: A] :
                    ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Xa)),R))
                   => pp(aa(A,bool,P5,Y3)) )
               => pp(aa(A,bool,P5,Xa)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B6))
               => pp(aa(A,bool,P5,X3)) ) ) )
       => wf(A,R) ) ) ).

% wfI
tff(fact_6027_fst__image__Sigma,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: fun(A,set(B))] : image2(product_prod(A,B),A,product_fst(A,B),product_Sigma(A,B,A6,B6)) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_acw(set(A),fun(fun(A,set(B)),fun(A,bool)),A6),B6)) ).

% fst_image_Sigma
tff(fact_6028_refl__onI,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A6,aTP_Lamp_acv(set(A),fun(A,set(A)),A6))))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3)),R)) )
       => refl_on(A,A6,R) ) ) ).

% refl_onI
tff(fact_6029_refl__on__def,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
      ( refl_on(A,A6,R)
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A6,aTP_Lamp_acv(set(A),fun(A,set(A)),A6))))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4)),R)) ) ) ) ).

% refl_on_def
tff(fact_6030_UN__Times__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,E5: fun(C,set(A)),F4: fun(D,set(B)),A6: set(C),B6: set(D)] : aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image2(product_prod(C,D),set(product_prod(A,B)),aa(fun(C,fun(D,set(product_prod(A,B)))),fun(product_prod(C,D),set(product_prod(A,B))),product_case_prod(C,D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_acy(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),E5),F4)),product_Sigma(C,D,A6,aTP_Lamp_acz(set(D),fun(C,set(D)),B6)))) = product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(C,set(A),E5,A6)),aa(set(D),fun(A,set(B)),aTP_Lamp_ada(fun(D,set(B)),fun(set(D),fun(A,set(B))),F4),B6)) ).

% UN_Times_distrib
tff(fact_6031_swap__product,axiom,
    ! [A: $tType,B: $tType,A6: set(B),B6: set(A)] : image2(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_mv(B,fun(A,product_prod(A,B)))),product_Sigma(B,A,A6,aTP_Lamp_acm(set(A),fun(B,set(A)),B6))) = product_Sigma(A,B,B6,aTP_Lamp_aci(set(B),fun(A,set(B)),A6)) ).

% swap_product
tff(fact_6032_card__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B6)) ).

% card_cartesian_product
tff(fact_6033_map__prod__surj__on,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F3: fun(B,A),A6: set(B),A10: set(A),G3: fun(D,C),B6: set(D),B12: set(C)] :
      ( ( image2(B,A,F3,A6) = A10 )
     => ( ( image2(D,C,G3,B6) = B12 )
       => ( image2(product_prod(B,D),product_prod(A,C),product_map_prod(B,A,D,C,F3,G3),product_Sigma(B,D,A6,aTP_Lamp_adb(set(D),fun(B,set(D)),B6))) = product_Sigma(A,C,A10,aTP_Lamp_adc(set(C),fun(A,set(C)),B12)) ) ) ) ).

% map_prod_surj_on
tff(fact_6034_map__prod__inj__on,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F3: fun(A,B),A6: set(A),G3: fun(C,D),B6: set(C)] :
      ( inj_on(A,B,F3,A6)
     => ( inj_on(C,D,G3,B6)
       => inj_on(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F3,G3),product_Sigma(A,C,A6,aTP_Lamp_adc(set(C),fun(A,set(C)),B6))) ) ) ).

% map_prod_inj_on
tff(fact_6035_wo__rel_Ocases__Total,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))),field2(A,R)))
       => ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
         => ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R))
             => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_6036_natLeq__on__wo__rel,axiom,
    ! [N: nat] : bNF_Wellorder_wo_rel(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aaw(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_wo_rel
tff(fact_6037_wo__rel_Omax2__greater__among,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R)))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R,A3,B2))),R))
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R,A3,B2))),R))
            & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),bNF_We1388413361240627857o_max2(A,R,A3,B2)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_6038_snd__image__Sigma,axiom,
    ! [A: $tType,B: $tType,A6: set(B),B6: fun(B,set(A))] : image2(product_prod(B,A),A,product_snd(B,A),product_Sigma(B,A,A6,B6)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(B,set(A),B6,A6)) ).

% snd_image_Sigma
tff(fact_6039_subset__fst__snd,axiom,
    ! [B: $tType,A: $tType,A6: set(product_prod(A,B))] : pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),product_Sigma(A,B,image2(product_prod(A,B),A,product_fst(A,B),A6),aTP_Lamp_add(set(product_prod(A,B)),fun(A,set(B)),A6)))) ).

% subset_fst_snd
tff(fact_6040_image__paired__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F3: fun(C,A),G3: fun(D,B),A6: set(C),B6: set(D)] : image2(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_ma(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F3),G3)),product_Sigma(C,D,A6,aTP_Lamp_acz(set(D),fun(C,set(D)),B6))) = product_Sigma(A,B,image2(C,A,F3,A6),aa(set(D),fun(A,set(B)),aTP_Lamp_ade(fun(D,B),fun(set(D),fun(A,set(B))),G3),B6)) ).

% image_paired_Times
tff(fact_6041_uniformly__continuous__on__uniformity,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [S2: set(A),F3: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,S2,F3)
        <=> filterlim(product_prod(A,A),product_prod(B,B),aa(fun(A,fun(A,product_prod(B,B))),fun(product_prod(A,A),product_prod(B,B)),product_case_prod(A,A,product_prod(B,B)),aTP_Lamp_adf(fun(A,B),fun(A,fun(A,product_prod(B,B))),F3)),topolo7806501430040627800ormity(B),aa(filter(product_prod(A,A)),filter(product_prod(A,A)),aa(filter(product_prod(A,A)),fun(filter(product_prod(A,A)),filter(product_prod(A,A))),inf_inf(filter(product_prod(A,A))),topolo7806501430040627800ormity(A)),principal(product_prod(A,A),product_Sigma(A,A,S2,aTP_Lamp_adg(set(A),fun(A,set(A)),S2))))) ) ) ).

% uniformly_continuous_on_uniformity
tff(fact_6042_lists__length__Suc__eq,axiom,
    ! [A: $tType,A6: set(A),N: nat] : aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_adh(set(A),fun(nat,fun(list(A),bool)),A6),N)) = image2(product_prod(list(A),A),list(A),aa(fun(list(A),fun(A,list(A))),fun(product_prod(list(A),A),list(A)),product_case_prod(list(A),A,list(A)),aTP_Lamp_mo(list(A),fun(A,list(A)))),product_Sigma(list(A),A,aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_aj(set(A),fun(nat,fun(list(A),bool)),A6),N)),aTP_Lamp_adi(set(A),fun(list(A),set(A)),A6))) ).

% lists_length_Suc_eq
tff(fact_6043_pairs__le__eq__Sigma,axiom,
    ! [M2: nat] : aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ib(nat,fun(nat,fun(nat,bool)),M2))) = product_Sigma(nat,nat,aa(nat,set(nat),set_ord_atMost(nat),M2),aTP_Lamp_adj(nat,fun(nat,set(nat)),M2)) ).

% pairs_le_eq_Sigma
tff(fact_6044_Sigma__def,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: fun(A,set(B))] : product_Sigma(A,B,A6,B6) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image2(A,set(product_prod(A,B)),aTP_Lamp_adl(fun(A,set(B)),fun(A,set(product_prod(A,B))),B6),A6)) ).

% Sigma_def
tff(fact_6045_product__fold,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(B),bool,finite_finite2(B),B6))
       => ( product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_adn(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B6),bot_bot(set(product_prod(A,B))),A6) ) ) ) ).

% product_fold
tff(fact_6046_well__order__induct__imp,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P2: fun(A,bool),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( ! [X3: A] :
            ( ! [Y4: A] :
                ( ( ( Y4 != X3 )
                  & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3)),R)) )
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),field2(A,R)))
                 => pp(aa(A,bool,P2,Y4)) ) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),field2(A,R)))
             => pp(aa(A,bool,P2,X3)) ) )
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field2(A,R)))
         => pp(aa(A,bool,P2,A3)) ) ) ) ).

% well_order_induct_imp
tff(fact_6047_wo__rel_Ominim__least,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B6: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),B6))
         => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We6954850376910717587_minim(A,R,B6)),B2)),R)) ) ) ) ).

% wo_rel.minim_least
tff(fact_6048_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R: A,S2: B,R2: set(product_prod(A,B)),S7: B] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R),S2)),R2))
     => ( ( S7 = S2 )
       => pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R),S7)),R2)) ) ) ).

% ssubst_Pair_rhs
tff(fact_6049_wo__rel_Oequals__minim,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B6: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),B6))
         => ( ! [B4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B6))
               => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B4)),R)) )
           => ( A3 = bNF_We6954850376910717587_minim(A,R,B6) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_6050_mult__inj__if__coprime__nat,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),A6: set(A),G3: fun(B,nat),B6: set(B)] :
      ( inj_on(A,nat,F3,A6)
     => ( inj_on(B,nat,G3,B6)
       => ( ! [A5: A,B4: B] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A6))
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B4),B6))
               => algebr8660921524188924756oprime(nat,aa(A,nat,F3,A5),aa(B,nat,G3,B4)) ) )
         => inj_on(product_prod(A,B),nat,aa(fun(A,fun(B,nat)),fun(product_prod(A,B),nat),product_case_prod(A,B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_ado(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),F3),G3)),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6))) ) ) ) ).

% mult_inj_if_coprime_nat
tff(fact_6051_image__split__eq__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(C,A),G3: fun(C,B),A6: set(C)] : image2(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_acc(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F3),G3),A6) = product_Sigma(A,B,image2(C,A,F3,A6),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_adp(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F3),G3),A6)) ).

% image_split_eq_Sigma
tff(fact_6052_coprime__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C3: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
        <=> ( algebr8660921524188924756oprime(A,C3,A3)
            & algebr8660921524188924756oprime(A,C3,B2) ) ) ) ).

% coprime_mult_right_iff
tff(fact_6053_coprime__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C3: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3)
        <=> ( algebr8660921524188924756oprime(A,A3,C3)
            & algebr8660921524188924756oprime(A,B2,C3) ) ) ) ).

% coprime_mult_left_iff
tff(fact_6054_coprime__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,A3)
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).

% coprime_self
tff(fact_6055_coprime__power__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,N: nat] :
          ( algebr8660921524188924756oprime(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))
        <=> ( algebr8660921524188924756oprime(A,A3,B2)
            | ( N = zero_zero(nat) ) ) ) ) ).

% coprime_power_right_iff
tff(fact_6056_coprime__power__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,N: nat,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),B2)
        <=> ( algebr8660921524188924756oprime(A,A3,B2)
            | ( N = zero_zero(nat) ) ) ) ) ).

% coprime_power_left_iff
tff(fact_6057_coprime__mod__right__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,A3,modulo_modulo(A,B2,A3))
          <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mod_right_iff
tff(fact_6058_coprime__mod__left__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,modulo_modulo(A,A3,B2),B2)
          <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mod_left_iff
tff(fact_6059_coprime__0__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,zero_zero(A))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).

% coprime_0_right_iff
tff(fact_6060_coprime__0__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,zero_zero(A),A3)
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).

% coprime_0_left_iff
tff(fact_6061_coprime__mult__self__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A)))
            & algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mult_self_left_iff
tff(fact_6062_coprime__mult__self__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A)))
            & algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mult_self_right_iff
tff(fact_6063_gcd__mult__left__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C3: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B2,C3)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_left_left_cancel
tff(fact_6064_gcd__mult__left__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C3: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B2,C3)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_left_right_cancel
tff(fact_6065_gcd__mult__right__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C3)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_right_left_cancel
tff(fact_6066_gcd__mult__right__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C3)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_mult_right_right_cancel
tff(fact_6067_finite__vimage__Suc__iff,axiom,
    ! [F4: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),vimage(nat,nat,suc,F4)))
    <=> pp(aa(set(nat),bool,finite_finite2(nat),F4)) ) ).

% finite_vimage_Suc_iff
tff(fact_6068_coprime__Suc__0__right,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,zero_zero(nat))) ).

% coprime_Suc_0_right
tff(fact_6069_coprime__Suc__0__left,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,zero_zero(nat)),N) ).

% coprime_Suc_0_left
tff(fact_6070_coprime__divisors,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C3: A,B2: A,D3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),D3))
           => ( algebr8660921524188924756oprime(A,C3,D3)
             => algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% coprime_divisors
tff(fact_6071_coprime__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B2,A3)
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_commute
tff(fact_6072_coprime__Suc__left__nat,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,N),N) ).

% coprime_Suc_left_nat
tff(fact_6073_coprime__Suc__right__nat,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,N)) ).

% coprime_Suc_right_nat
tff(fact_6074_coprime__1__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,one_one(A)) ) ).

% coprime_1_right
tff(fact_6075_coprime__1__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,one_one(A),A3) ) ).

% coprime_1_left
tff(fact_6076_coprime__crossproduct__nat,axiom,
    ! [A3: nat,D3: nat,B2: nat,C3: nat] :
      ( algebr8660921524188924756oprime(nat,A3,D3)
     => ( algebr8660921524188924756oprime(nat,B2,C3)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),C3) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),D3) )
        <=> ( ( A3 = B2 )
            & ( C3 = D3 ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_6077_mult__mod__cancel__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [A3: A,N: A,M2: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),N),M2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),N),M2) )
         => ( algebr8660921524188924756oprime(A,M2,N)
           => ( modulo_modulo(A,A3,M2) = modulo_modulo(A,B2,M2) ) ) ) ) ).

% mult_mod_cancel_right
tff(fact_6078_mult__mod__cancel__left,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [N: A,A3: A,M2: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N),A3),M2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N),B2),M2) )
         => ( algebr8660921524188924756oprime(A,M2,N)
           => ( modulo_modulo(A,A3,M2) = modulo_modulo(A,B2,M2) ) ) ) ) ).

% mult_mod_cancel_left
tff(fact_6079_is__unit__right__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_right_imp_coprime
tff(fact_6080_is__unit__left__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_left_imp_coprime
tff(fact_6081_coprime__common__divisor,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C3: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A))) ) ) ) ) ).

% coprime_common_divisor
tff(fact_6082_coprime__absorb__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y),X))
         => ( algebr8660921524188924756oprime(A,X,Y)
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y),one_one(A))) ) ) ) ).

% coprime_absorb_right
tff(fact_6083_coprime__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,D3: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C3,D3)
         => ( ! [E2: A] :
                ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),one_one(A)))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),A3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),B2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),C3)) ) ) )
           => ( ! [E2: A] :
                  ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),one_one(A)))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),B2))
                     => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),D3)) ) ) )
             => algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% coprime_imp_coprime
tff(fact_6084_coprime__absorb__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y))
         => ( algebr8660921524188924756oprime(A,X,Y)
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),one_one(A))) ) ) ) ).

% coprime_absorb_left
tff(fact_6085_not__coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
           => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A)))
             => ~ algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% not_coprimeI
tff(fact_6086_not__coprimeE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ~ algebr8660921524188924756oprime(A,A3,B2)
         => ~ ! [C2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A))) ) ) ) ) ).

% not_coprimeE
tff(fact_6087_coprime__def,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
        <=> ! [C4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),B2))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),one_one(A))) ) ) ) ) ).

% coprime_def
tff(fact_6088_coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ! [C2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A))) ) )
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprimeI
tff(fact_6089_coprime__dvd__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% coprime_dvd_mult_right_iff
tff(fact_6090_coprime__dvd__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,C3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% coprime_dvd_mult_left_iff
tff(fact_6091_divides__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C3))
           => ( algebr8660921524188924756oprime(A,A3,B2)
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C3)) ) ) ) ) ).

% divides_mult
tff(fact_6092_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,A6: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,N)),A6)) = aa(set(nat),set(nat),insert(nat,N),vimage(nat,nat,suc,A6)) ).

% vimage_Suc_insert_Suc
tff(fact_6093_vimage__Times,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,product_prod(B,C)),A6: set(B),B6: set(C)] : vimage(A,product_prod(B,C),F3,product_Sigma(B,C,A6,aTP_Lamp_adq(set(C),fun(B,set(C)),B6))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),F3),A6)),vimage(A,C,aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),F3),B6)) ).

% vimage_Times
tff(fact_6094_Pair__vimage__Sigma,axiom,
    ! [B: $tType,A: $tType,X: B,A6: set(B),F3: fun(B,set(A))] :
      ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A6))
       => ( vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),product_Sigma(B,A,A6,F3)) = aa(B,set(A),F3,X) ) )
      & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A6))
       => ( vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),product_Sigma(B,A,A6,F3)) = bot_bot(set(A)) ) ) ) ).

% Pair_vimage_Sigma
tff(fact_6095_vimage__Suc__insert__0,axiom,
    ! [A6: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),insert(nat,zero_zero(nat)),A6)) = vimage(nat,nat,suc,A6) ).

% vimage_Suc_insert_0
tff(fact_6096_invertible__coprime,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B2: A,C3: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C3) = one_one(A) )
         => algebr8660921524188924756oprime(A,A3,C3) ) ) ).

% invertible_coprime
tff(fact_6097_gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,A4: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) != zero_zero(A) )
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
           => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
             => algebr8660921524188924756oprime(A,A4,B3) ) ) ) ) ).

% gcd_coprime
tff(fact_6098_gcd__coprime__exists,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) != zero_zero(A) )
         => ? [A14: A,B10: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A14),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
              & ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B10),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
              & algebr8660921524188924756oprime(A,A14,B10) ) ) ) ).

% gcd_coprime_exists
tff(fact_6099_div__gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( ( A3 != zero_zero(A) )
            | ( B2 != zero_zero(A) ) )
         => algebr8660921524188924756oprime(A,divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),divide_divide(A,B2,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ) ).

% div_gcd_coprime
tff(fact_6100_vimage__fst,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] : vimage(product_prod(A,B),A,product_fst(A,B),A6) = product_Sigma(A,B,A6,aTP_Lamp_ack(A,set(B))) ).

% vimage_fst
tff(fact_6101_vimage__snd,axiom,
    ! [A: $tType,B: $tType,A6: set(B)] : vimage(product_prod(A,B),B,product_snd(A,B),A6) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_aci(set(B),fun(A,set(B)),A6)) ).

% vimage_snd
tff(fact_6102_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => algebr8660921524188924756oprime(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),N) ) ).

% coprime_diff_one_left_nat
tff(fact_6103_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => algebr8660921524188924756oprime(nat,N,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_6104_card__vimage__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),D5: set(A),A6: set(B)] :
      ( inj_on(A,B,F3,D5)
     => ( pp(aa(set(B),bool,finite_finite2(B),A6))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F3,A6)),D5))),aa(set(B),nat,finite_card(B),A6))) ) ) ).

% card_vimage_inj_on_le
tff(fact_6105_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),field_char_0_Rats(real)))
     => ~ ! [M: nat,N3: nat] :
            ( ( N3 != zero_zero(nat) )
           => ( ( aa(real,real,abs_abs(real),X) = divide_divide(real,aa(nat,real,semiring_1_of_nat(real),M),aa(nat,real,semiring_1_of_nat(real),N3)) )
             => ~ algebr8660921524188924756oprime(nat,M,N3) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_6106_set__decode__div__2,axiom,
    ! [X: nat] : nat_set_decode(divide_divide(nat,X,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = vimage(nat,nat,suc,nat_set_decode(X)) ).

% set_decode_div_2
tff(fact_6107_set__encode__vimage__Suc,axiom,
    ! [A6: set(nat)] : aa(set(nat),nat,nat_set_encode,vimage(nat,nat,suc,A6)) = divide_divide(nat,aa(set(nat),nat,nat_set_encode,A6),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% set_encode_vimage_Suc
tff(fact_6108_Restr__natLeq,axiom,
    ! [N: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_bt(nat,fun(nat,bool)),N)),aTP_Lamp_adr(nat,fun(nat,set(nat)),N))) = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aaw(nat,fun(nat,fun(nat,bool)),N))) ).

% Restr_natLeq
tff(fact_6109_sum__mult__sum__if__inj,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [F3: fun(A,B),G3: fun(C,B),A6: set(A),B6: set(C)] :
          ( inj_on(product_prod(A,C),B,aa(fun(A,fun(C,B)),fun(product_prod(A,C),B),product_case_prod(A,C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_bd(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),F3),G3)),product_Sigma(A,C,A6,aTP_Lamp_adc(set(C),fun(A,set(C)),B6)))
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A6)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G3),B6)) = aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7311177749621191930dd_sum(B,B),id(B)),aa(fun(B,bool),set(B),collect(B),aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_ads(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),F3),G3),A6),B6))) ) ) ) ).

% sum_mult_sum_if_inj
tff(fact_6110_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_6111_case__prod__Pair,axiom,
    ! [B: $tType,A: $tType] : aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)) = id(product_prod(A,B)) ).

% case_prod_Pair
tff(fact_6112_id__funpow,axiom,
    ! [A: $tType,N: nat] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),id(A)) = id(A) ).

% id_funpow
tff(fact_6113_rotate0,axiom,
    ! [A: $tType] : rotate(A,zero_zero(nat)) = id(list(A)) ).

% rotate0
tff(fact_6114_apfst__id,axiom,
    ! [B: $tType,A: $tType] : product_apfst(A,A,B,id(A)) = id(product_prod(A,B)) ).

% apfst_id
tff(fact_6115_apsnd__id,axiom,
    ! [B: $tType,A: $tType] : aa(fun(B,B),fun(product_prod(A,B),product_prod(A,B)),product_apsnd(B,B,A),id(B)) = id(product_prod(A,B)) ).

% apsnd_id
tff(fact_6116_push__bit__0__id,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4730199178511100633sh_bit(A,zero_zero(nat)) = id(A) ) ) ).

% push_bit_0_id
tff(fact_6117_comp__the__Some,axiom,
    ! [A: $tType] : aa(fun(A,option(A)),fun(A,A),comp(option(A),A,A,the2(A)),some(A)) = id(A) ).

% comp_the_Some
tff(fact_6118_drop__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4197421643247451524op_bit(A,zero_zero(nat)) = id(A) ) ) ).

% drop_bit_0
tff(fact_6119_coprime__crossproduct__int,axiom,
    ! [A3: int,D3: int,B2: int,C3: int] :
      ( algebr8660921524188924756oprime(int,A3,D3)
     => ( algebr8660921524188924756oprime(int,B2,C3)
       => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),A3)),aa(int,int,abs_abs(int),C3)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),B2)),aa(int,int,abs_abs(int),D3)) )
        <=> ( ( aa(int,int,abs_abs(int),A3) = aa(int,int,abs_abs(int),B2) )
            & ( aa(int,int,abs_abs(int),C3) = aa(int,int,abs_abs(int),D3) ) ) ) ) ) ).

% coprime_crossproduct_int
tff(fact_6120_map__prod_Oidentity,axiom,
    ! [B: $tType,A: $tType] : product_map_prod(A,A,B,B,aTP_Lamp_abo(A,A),aTP_Lamp_abp(B,B)) = id(product_prod(A,B)) ).

% map_prod.identity
tff(fact_6121_funpow__simps__right_I1_J,axiom,
    ! [A: $tType,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F3) = id(A) ).

% funpow_simps_right(1)
tff(fact_6122_natLeq__Linear__order,axiom,
    order_679001287576687338der_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Linear_order
tff(fact_6123_natLeq__Total,axiom,
    total_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Total
tff(fact_6124_natLeq__Refl,axiom,
    refl_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Refl
tff(fact_6125_apfst__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,C)] : product_apfst(A,C,B,F3) = product_map_prod(A,C,B,B,F3,id(B)) ).

% apfst_def
tff(fact_6126_apsnd__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] : aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3) = product_map_prod(A,A,B,C,id(A),F3) ).

% apsnd_def
tff(fact_6127_natLeq__natLess__Id,axiom,
    bNF_Ca8459412986667044542atLess = aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),minus_minus(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),id2(nat)) ).

% natLeq_natLess_Id
tff(fact_6128_Field__natLeq,axiom,
    field2(nat,bNF_Ca8665028551170535155natLeq) = top_top(set(nat)) ).

% Field_natLeq
tff(fact_6129_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less_eq(nat))) ).

% natLeq_def
tff(fact_6130_fst__diag__id,axiom,
    ! [A: $tType,Z2: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),comp(product_prod(A,A),A,A,product_fst(A,A)),aTP_Lamp_mq(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).

% fst_diag_id
tff(fact_6131_snd__diag__id,axiom,
    ! [A: $tType,Z2: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),comp(product_prod(A,A),A,A,product_snd(A,A)),aTP_Lamp_mq(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).

% snd_diag_id
tff(fact_6132_Restr__natLeq2,axiom,
    ! [N: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,order_underS(nat,bNF_Ca8665028551170535155natLeq,N),aTP_Lamp_adt(nat,fun(nat,set(nat)),N))) = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aaw(nat,fun(nat,fun(nat,bool)),N))) ).

% Restr_natLeq2
tff(fact_6133_relation__of__def,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),A6: set(A)] : order_relation_of(A,P2,A6) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_adu(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),P2),A6))) ).

% relation_of_def
tff(fact_6134_underS__I,axiom,
    ! [A: $tType,I2: A,J2: A,R2: set(product_prod(A,A))] :
      ( ( I2 != J2 )
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2)),R2))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),order_underS(A,R2,J2))) ) ) ).

% underS_I
tff(fact_6135_underS__E,axiom,
    ! [A: $tType,I2: A,R2: set(product_prod(A,A)),J2: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),order_underS(A,R2,J2)))
     => ( ( I2 != J2 )
        & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J2)),R2)) ) ) ).

% underS_E
tff(fact_6136_underS__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_underS(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_adv(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).

% underS_def
tff(fact_6137_natLeq__underS__less,axiom,
    ! [N: nat] : order_underS(nat,bNF_Ca8665028551170535155natLeq,N) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_bt(nat,fun(nat,bool)),N)) ).

% natLeq_underS_less
tff(fact_6138_underS__incl__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R,A3)),order_underS(A,R,B2)))
          <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ) ) ).

% underS_incl_iff
tff(fact_6139_swap__comp__swap,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),product_prod(A,B)),comp(product_prod(B,A),product_prod(A,B),product_prod(A,B),product_swap(B,A)),product_swap(A,B)) = id(product_prod(A,B)) ).

% swap_comp_swap
tff(fact_6140_sorted__insort__insert__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F3,Xs))
         => sorted_wrt(A,ord_less_eq(A),map(B,A,F3,linord329482645794927042rt_key(B,A,F3,X,Xs))) ) ) ).

% sorted_insort_insert_key
tff(fact_6141_swap__swap,axiom,
    ! [B: $tType,A: $tType,P: product_prod(A,B)] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P)) = P ).

% swap_swap
tff(fact_6142_swap__simp,axiom,
    ! [A: $tType,B: $tType,X: B,Y: A] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X) ).

% swap_simp
tff(fact_6143_case__swap,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,fun(B,A)),P: product_prod(C,B)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aTP_Lamp_adw(fun(C,fun(B,A)),fun(B,fun(C,A)),F3)),aa(product_prod(C,B),product_prod(B,C),product_swap(C,B),P)) = aa(product_prod(C,B),A,aa(fun(C,fun(B,A)),fun(product_prod(C,B),A),product_case_prod(C,B,A),F3),P) ).

% case_swap
tff(fact_6144_fst__swap,axiom,
    ! [A: $tType,B: $tType,X: product_prod(B,A)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),X)) = aa(product_prod(B,A),A,product_snd(B,A),X) ).

% fst_swap
tff(fact_6145_snd__swap,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),X)) = aa(product_prod(A,B),A,product_fst(A,B),X) ).

% snd_swap
tff(fact_6146_pair__in__swap__image,axiom,
    ! [A: $tType,B: $tType,Y: A,X: B,A6: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A),A6)))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)),A6)) ) ).

% pair_in_swap_image
tff(fact_6147_surj__swap,axiom,
    ! [B: $tType,A: $tType] : image2(product_prod(B,A),product_prod(A,B),product_swap(B,A),top_top(set(product_prod(B,A)))) = top_top(set(product_prod(A,B))) ).

% surj_swap
tff(fact_6148_bij__swap,axiom,
    ! [A: $tType,B: $tType] : bij_betw(product_prod(A,B),product_prod(B,A),product_swap(A,B),top_top(set(product_prod(A,B))),top_top(set(product_prod(B,A)))) ).

% bij_swap
tff(fact_6149_inj__swap,axiom,
    ! [B: $tType,A: $tType,A6: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),product_swap(A,B),A6) ).

% inj_swap
tff(fact_6150_product__swap,axiom,
    ! [A: $tType,B: $tType,A6: set(B),B6: set(A)] : image2(product_prod(B,A),product_prod(A,B),product_swap(B,A),product_Sigma(B,A,A6,aTP_Lamp_acm(set(A),fun(B,set(A)),B6))) = product_Sigma(A,B,B6,aTP_Lamp_aci(set(B),fun(A,set(B)),A6)) ).

% product_swap
tff(fact_6151_sorted__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),X: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),linord329482645794927042rt_key(A,A,aTP_Lamp_ni(A,A),X,Xs)) ) ) ).

% sorted_insort_insert
tff(fact_6152_prod_Oswap__def,axiom,
    ! [B: $tType,A: $tType,P: product_prod(A,B)] : aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(product_prod(A,B),B,product_snd(A,B),P)),aa(product_prod(A,B),A,product_fst(A,B),P)) ).

% prod.swap_def
tff(fact_6153_VEBT_Osimps_I7_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),F22: fun(bool,fun(bool,A)),X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_rec_VEBT(A),F1),F22),vEBT_Node(X11,X12,X13,X14)) = aa(A,A,aa(vEBT_VEBT,fun(A,A),aa(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A)),aa(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A)))),F1,X11),X12),map(vEBT_VEBT,product_prod(vEBT_VEBT,A),aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A)),aTP_Lamp_adx(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A))),F1),F22),X13)),X14),aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_rec_VEBT(A),F1),F22),X14)) ).

% VEBT.simps(7)
tff(fact_6154_pair__lessI2,axiom,
    ! [A3: nat,B2: nat,S2: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S2),T2))
       => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_less)) ) ) ).

% pair_lessI2
tff(fact_6155_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Z2))),fun_pair_less))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),Z2)) ) ).

% pair_less_iff1
tff(fact_6156_VEBT_Osimps_I8_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),F22: fun(bool,fun(bool,A)),X21: bool,X22: bool] : aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_rec_VEBT(A),F1),F22),vEBT_Leaf(X21,X22)) = aa(bool,A,aa(bool,fun(bool,A),F22,X21),X22) ).

% VEBT.simps(8)
tff(fact_6157_pair__lessI1,axiom,
    ! [A3: nat,B2: nat,S2: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_less)) ) ).

% pair_lessI1
tff(fact_6158_pair__leqI2,axiom,
    ! [A3: nat,B2: nat,S2: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),S2),T2))
       => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_leq)) ) ) ).

% pair_leqI2
tff(fact_6159_pair__leqI1,axiom,
    ! [A3: nat,B2: nat,S2: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_leq)) ) ).

% pair_leqI1
tff(fact_6160_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_ady(A,fun(A,bool)),aTP_Lamp_adz(A,fun(A,bool)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_6161_length__upto,axiom,
    ! [I2: int,J2: int] : aa(list(int),nat,size_size(list(int)),upto(I2,J2)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),J2),I2)),one_one(int))) ).

% length_upto
tff(fact_6162_ordering__top_Oextremum__uniqueI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A3))
       => ( A3 = Top ) ) ) ).

% ordering_top.extremum_uniqueI
tff(fact_6163_ordering__top_Onot__eq__extremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( ( A3 != Top )
      <=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),Top)) ) ) ).

% ordering_top.not_eq_extremum
tff(fact_6164_ordering__top_Oextremum__unique,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A3))
      <=> ( A3 = Top ) ) ) ).

% ordering_top.extremum_unique
tff(fact_6165_ordering__top_Oextremum__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Top),A3)) ) ).

% ordering_top.extremum_strict
tff(fact_6166_ordering__top_Oextremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),Top)) ) ).

% ordering_top.extremum
tff(fact_6167_gcd__nat_Oordering__top__axioms,axiom,
    ordering_top(nat,dvd_dvd(nat),aTP_Lamp_aax(nat,fun(nat,bool)),zero_zero(nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_6168_top_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ordering_top(A,ord_less_eq(A),ord_less(A),top_top(A)) ) ).

% top.ordering_top_axioms
tff(fact_6169_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_ac(nat,fun(nat,bool)),aTP_Lamp_bt(nat,fun(nat,bool)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_6170_ratrel__iff,axiom,
    ! [X: product_prod(int,int),Y: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),Y))
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Y) != zero_zero(int) )
        & ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Y)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Y)),aa(product_prod(int,int),int,product_snd(int,int),X)) ) ) ) ).

% ratrel_iff
tff(fact_6171_dropWhile__nth,axiom,
    ! [A: $tType,J2: nat,P2: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),aa(list(A),nat,size_size(list(A)),dropWhile(A,P2,Xs))))
     => ( aa(nat,A,nth(A,dropWhile(A,P2,Xs)),J2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)))) ) ) ).

% dropWhile_nth
tff(fact_6172_length__dropWhile__le,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),dropWhile(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_dropWhile_le
tff(fact_6173_sorted__dropWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),dropWhile(A,P2,Xs)) ) ) ).

% sorted_dropWhile
tff(fact_6174_dropWhile__eq__drop,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : dropWhile(A,P2,Xs) = drop(A,aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)),Xs) ).

% dropWhile_eq_drop
tff(fact_6175_ratrel__def,axiom,
    ! [X5: product_prod(int,int),Xa: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X5),Xa))
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X5) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Xa) != zero_zero(int) )
        & ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X5)),aa(product_prod(int,int),int,product_snd(int,int),Xa)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X5)) ) ) ) ).

% ratrel_def
tff(fact_6176_plus__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod(int,int),X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa2),Xa2))
     => ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(product_prod(int,int),rat,abs_Rat,Xa2)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa2)),aa(product_prod(int,int),int,product_snd(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa2)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_6177_times__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod(int,int),X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa2),Xa2))
     => ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
       => ( aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(product_prod(int,int),rat,abs_Rat,Xa2)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa2)),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa2)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_6178_Rat_Opositive_Oabs__eq,axiom,
    ! [X: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
     => ( pp(aa(rat,bool,positive,aa(product_prod(int,int),rat,abs_Rat,X)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ).

% Rat.positive.abs_eq
tff(fact_6179_find__dropWhile,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : find(A,P2,Xs) = case_list(option(A),A,none(A),aTP_Lamp_aea(A,fun(list(A),option(A))),dropWhile(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P2),Xs)) ).

% find_dropWhile
tff(fact_6180_Rat_Opositive__mult,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa(rat,bool,positive,X))
     => ( pp(aa(rat,bool,positive,Y))
       => pp(aa(rat,bool,positive,aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),X),Y))) ) ) ).

% Rat.positive_mult
tff(fact_6181_Rat_Opositive_Orep__eq,axiom,
    ! [X: rat] :
      ( pp(aa(rat,bool,positive,X))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),aa(rat,product_prod(int,int),rep_Rat,X))),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,X))))) ) ).

% Rat.positive.rep_eq
tff(fact_6182_Rat_Opositive_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(product_prod(int,int),bool),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),bool,bool,ratrel,fequal(bool)),aTP_Lamp_aeb(product_prod(int,int),bool)),aTP_Lamp_aeb(product_prod(int,int),bool))) ).

% Rat.positive.rsp
tff(fact_6183_transfer__rule__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1(B)
        & ring_1(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),plus_plus(A)),plus_plus(B)))
             => ( pp(aa(fun(B,B),bool,aa(fun(A,A),fun(fun(B,B),bool),bNF_rel_fun(A,B,A,B,R2,R2),uminus_uminus(A)),uminus_uminus(B)))
               => pp(aa(fun(int,B),bool,aa(fun(int,A),fun(fun(int,B),bool),bNF_rel_fun(int,int,A,B,fequal(int),R2),ring_1_of_int(A)),ring_1_of_int(B))) ) ) ) ) ) ).

% transfer_rule_of_int
tff(fact_6184_transfer__rule__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & semiring_numeral(B)
        & monoid_add(A)
        & semiring_numeral(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),plus_plus(A)),plus_plus(B)))
             => pp(aa(fun(num,B),bool,aa(fun(num,A),fun(fun(num,B),bool),bNF_rel_fun(num,num,A,B,fequal(num),R2),numeral_numeral(A)),numeral_numeral(B))) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_6185_power__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & power(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),times_times(A)),times_times(B)))
           => pp(aa(fun(B,fun(nat,B)),bool,aa(fun(A,fun(nat,A)),fun(fun(B,fun(nat,B)),bool),bNF_rel_fun(A,B,fun(nat,A),fun(nat,B),R2,bNF_rel_fun(nat,nat,A,B,fequal(nat),R2)),power_power(A)),power_power(B))) ) ) ) ).

% power_transfer
tff(fact_6186_transfer__rule__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1(B)
        & semiring_1(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R2,bNF_rel_fun(A,B,A,B,R2,R2)),plus_plus(A)),plus_plus(B)))
             => pp(aa(fun(nat,B),bool,aa(fun(nat,A),fun(fun(nat,B),bool),bNF_rel_fun(nat,nat,A,B,fequal(nat),R2),semiring_1_of_nat(A)),semiring_1_of_nat(B))) ) ) ) ) ).

% transfer_rule_of_nat
tff(fact_6187_Rat_Opositive__def,axiom,
    positive = aa(fun(product_prod(int,int),bool),fun(rat,bool),map_fun(rat,product_prod(int,int),bool,bool,rep_Rat,id(bool)),aTP_Lamp_aeb(product_prod(int,int),bool)) ).

% Rat.positive_def
tff(fact_6188_less__eq__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(nat,fun(nat,bool)),bool),bNF_rel_fun(nat,nat,fun(nat,bool),fun(nat,bool),fequal(nat),bNF_rel_fun(nat,nat,bool,bool,fequal(nat),fequal(bool))),ord_less_eq(nat)),ord_less_eq(nat))) ).

% less_eq_natural.rsp
tff(fact_6189_Suc_Orsp,axiom,
    pp(aa(fun(nat,nat),bool,aa(fun(nat,nat),fun(fun(nat,nat),bool),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),suc),suc)) ).

% Suc.rsp
tff(fact_6190_times__integer_Orsp,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),times_times(int)),times_times(int))) ).

% times_integer.rsp
tff(fact_6191_times__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),times_times(nat)),times_times(nat))) ).

% times_natural.rsp
tff(fact_6192_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R2: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,one_one(A)),one_one(B)))
           => pp(aa(fun(bool,B),bool,aa(fun(bool,A),fun(fun(bool,B),bool),bNF_rel_fun(bool,bool,A,B,fequal(bool),R2),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B))) ) ) ) ).

% transfer_rule_of_bool
tff(fact_6193_times__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_aec(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_aec(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).

% times_rat.rsp
tff(fact_6194_plus__rat_Orsp,axiom,
    pp(aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_aed(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_aed(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).

% plus_rat.rsp
tff(fact_6195_plus__rat_Otransfer,axiom,
    pp(aa(fun(rat,fun(rat,rat)),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),bool),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_aed(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat))) ).

% plus_rat.transfer
tff(fact_6196_Rat_Opositive_Otransfer,axiom,
    pp(aa(fun(rat,bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(rat,bool),bool),bNF_rel_fun(product_prod(int,int),rat,bool,bool,pcr_rat,fequal(bool)),aTP_Lamp_aeb(product_prod(int,int),bool)),positive)) ).

% Rat.positive.transfer
tff(fact_6197_times__rat_Otransfer,axiom,
    pp(aa(fun(rat,fun(rat,rat)),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),bool),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_aec(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat))) ).

% times_rat.transfer
tff(fact_6198_times__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int))) ).

% times_int.transfer
tff(fact_6199_minus__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int))) ).

% minus_int.transfer
tff(fact_6200_zero__int_Otransfer,axiom,
    pp(aa(int,bool,aa(product_prod(nat,nat),fun(int,bool),pcr_int,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),zero_zero(int))) ).

% zero_int.transfer
tff(fact_6201_int__transfer,axiom,
    pp(aa(fun(nat,int),bool,aa(fun(nat,product_prod(nat,nat)),fun(fun(nat,int),bool),bNF_rel_fun(nat,nat,product_prod(nat,nat),int,fequal(nat),pcr_int),aTP_Lamp_aee(nat,product_prod(nat,nat))),semiring_1_of_nat(int))) ).

% int_transfer
tff(fact_6202_uminus__int_Otransfer,axiom,
    pp(aa(fun(int,int),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(int,int),bool),bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_kl(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int))) ).

% uminus_int.transfer
tff(fact_6203_one__int_Otransfer,axiom,
    pp(aa(int,bool,aa(product_prod(nat,nat),fun(int,bool),pcr_int,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),one_one(int))) ).

% one_int.transfer
tff(fact_6204_less__eq__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_kn(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less_eq(int))) ).

% less_eq_int.transfer
tff(fact_6205_plus__int_Otransfer,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int))) ).

% plus_int.transfer
tff(fact_6206_times__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% times_int.rsp
tff(fact_6207_minus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% minus_int.rsp
tff(fact_6208_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V3: nat] :
      ( pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),U),V3)))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),V3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y) ) ) ).

% intrel_iff
tff(fact_6209_zero__int_Orsp,axiom,
    pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat)))) ).

% zero_int.rsp
tff(fact_6210_uminus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),product_prod(nat,nat)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_kl(nat,fun(nat,product_prod(nat,nat))))),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_kl(nat,fun(nat,product_prod(nat,nat)))))) ).

% uminus_int.rsp
tff(fact_6211_one__int_Orsp,axiom,
    pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat)))) ).

% one_int.rsp
tff(fact_6212_less__eq__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_kn(nat,fun(nat,fun(product_prod(nat,nat),bool))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_kn(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_eq_int.rsp
tff(fact_6213_plus__int_Orsp,axiom,
    pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% plus_int.rsp
tff(fact_6214_horner__sum__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( comm_semiring_0(B)
        & comm_semiring_0(A) )
     => ! [A6: fun(A,fun(B,bool)),B6: fun(C,fun(D,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A6,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A6,bNF_rel_fun(A,B,A,B,A6,A6)),plus_plus(A)),plus_plus(B)))
           => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A6,bNF_rel_fun(A,B,A,B,A6,A6)),times_times(A)),times_times(B)))
             => pp(aa(fun(fun(D,B),fun(B,fun(list(D),B))),bool,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),bool),bNF_rel_fun(fun(C,A),fun(D,B),fun(A,fun(list(C),A)),fun(B,fun(list(D),B)),bNF_rel_fun(C,D,A,B,B6,A6),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A6,bNF_rel_fun(list(C),list(D),A,B,list_all2(C,D,B6),A6))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B))) ) ) ) ) ).

% horner_sum_transfer
tff(fact_6215_euclidean__size__times__nonunit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,B2)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_6216_euclidean__size__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A] :
          ( ( euclid6346220572633701492n_size(A,B2) = zero_zero(nat) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% euclidean_size_eq_0_iff
tff(fact_6217_size__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ( euclid6346220572633701492n_size(A,zero_zero(A)) = zero_zero(nat) ) ) ).

% size_0
tff(fact_6218_euclidean__size__greater__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),euclid6346220572633701492n_size(A,B2)))
        <=> ( B2 != zero_zero(A) ) ) ) ).

% euclidean_size_greater_0_iff
tff(fact_6219_length__transfer,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] : pp(aa(fun(list(B),nat),bool,aa(fun(list(A),nat),fun(fun(list(B),nat),bool),bNF_rel_fun(list(A),list(B),nat,nat,list_all2(A,B,A6),fequal(nat)),size_size(list(A))),size_size(list(B)))) ).

% length_transfer
tff(fact_6220_list__all2__lengthD,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),Ys))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% list_all2_lengthD
tff(fact_6221_list__all2__append2,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),Zs: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),append(B,Ys,Zs)))
    <=> ? [Us3: list(A),Vs3: list(A)] :
          ( ( Xs = append(A,Us3,Vs3) )
          & ( aa(list(A),nat,size_size(list(A)),Us3) = aa(list(B),nat,size_size(list(B)),Ys) )
          & ( aa(list(A),nat,size_size(list(A)),Vs3) = aa(list(B),nat,size_size(list(B)),Zs) )
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Us3),Ys))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Vs3),Zs)) ) ) ).

% list_all2_append2
tff(fact_6222_list__all2__append1,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),Xs: list(A),Ys: list(A),Zs: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),append(A,Xs,Ys)),Zs))
    <=> ? [Us3: list(B),Vs3: list(B)] :
          ( ( Zs = append(B,Us3,Vs3) )
          & ( aa(list(B),nat,size_size(list(B)),Us3) = aa(list(A),nat,size_size(list(A)),Xs) )
          & ( aa(list(B),nat,size_size(list(B)),Vs3) = aa(list(A),nat,size_size(list(A)),Ys) )
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),Us3))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Ys),Vs3)) ) ) ).

% list_all2_append1
tff(fact_6223_list__all2__append,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P2: fun(A,fun(B,bool)),Us: list(A),Vs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),append(A,Xs,Us)),append(B,Ys,Vs)))
      <=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),Ys))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Us),Vs)) ) ) ) ).

% list_all2_append
tff(fact_6224_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( euclid6346220572633701492n_size(A,A3) = euclid6346220572633701492n_size(A,B2) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
tff(fact_6225_euclidean__size__mult,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A] : euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2)) ) ).

% euclidean_size_mult
tff(fact_6226_list__all2__nthD,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),P: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),Ys))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(nat,A,nth(A,Xs),P)),aa(nat,B,nth(B,Ys),P))) ) ) ).

% list_all2_nthD
tff(fact_6227_list__all2__nthD2,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),P: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),Ys))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P),aa(list(B),nat,size_size(list(B)),Ys)))
       => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(nat,A,nth(A,Xs),P)),aa(nat,B,nth(B,Ys),P))) ) ) ).

% list_all2_nthD2
tff(fact_6228_list__all2__all__nthI,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B2: list(B),P2: fun(A,fun(B,bool))] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ! [N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),A3)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(nat,A,nth(A,A3),N3)),aa(nat,B,nth(B,B2),N3))) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),A3),B2)) ) ) ).

% list_all2_all_nthI
tff(fact_6229_list__all2__conv__all__nth,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),Ys))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys),I))) ) ) ) ).

% list_all2_conv_all_nth
tff(fact_6230_unit__iff__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
        <=> ( ( euclid6346220572633701492n_size(A,A3) = euclid6346220572633701492n_size(A,one_one(A)) )
            & ( A3 != zero_zero(A) ) ) ) ) ).

% unit_iff_euclidean_size
tff(fact_6231_size__mult__mono_H,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)))) ) ) ).

% size_mult_mono'
tff(fact_6232_size__mult__mono,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ).

% size_mult_mono
tff(fact_6233_euclidean__size__times__unit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = euclid6346220572633701492n_size(A,B2) ) ) ) ).

% euclidean_size_times_unit
tff(fact_6234_dvd__proper__imp__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( ( B2 != zero_zero(A) )
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2))) ) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_6235_dvd__imp__size__le,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( ( B2 != zero_zero(A) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2))) ) ) ) ).

% dvd_imp_size_le
tff(fact_6236_mod__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,modulo_modulo(A,A3,B2))),euclid6346220572633701492n_size(A,B2))) ) ) ).

% mod_size_less
tff(fact_6237_list__all2I,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B2: list(B),P2: fun(A,fun(B,bool))] :
      ( ! [X3: product_prod(A,B)] :
          ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X3),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,A3,B2))))
         => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2),X3)) )
     => ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),A3),B2)) ) ) ).

% list_all2I
tff(fact_6238_sum__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & monoid_add(A) )
     => ! [A6: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A6,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A6,bNF_rel_fun(A,B,A,B,A6,A6)),plus_plus(A)),plus_plus(B)))
           => pp(aa(fun(list(B),B),bool,aa(fun(list(A),A),fun(fun(list(B),B),bool),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A6),A6),groups8242544230860333062m_list(A)),groups8242544230860333062m_list(B))) ) ) ) ).

% sum_list_transfer
tff(fact_6239_list__all2__iff,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),Ys))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X4: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X4),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
           => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2),X4)) ) ) ) ).

% list_all2_iff
tff(fact_6240_divmod__cases,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,A3: A] :
          ( ( ( B2 != zero_zero(A) )
           => ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
             => ( A3 != aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B2)),B2) ) ) )
         => ( ( ( B2 != zero_zero(A) )
             => ! [Q3: A,R3: A] :
                  ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B2) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R3)),euclid6346220572633701492n_size(A,B2)))
                   => ( ( R3 != zero_zero(A) )
                     => ( ( divide_divide(A,A3,B2) = Q3 )
                       => ( ( modulo_modulo(A,A3,B2) = R3 )
                         => ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B2)),R3) ) ) ) ) ) ) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% divmod_cases
tff(fact_6241_prod__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_mult(B)
        & monoid_mult(A) )
     => ! [A6: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),A6,one_one(A)),one_one(B)))
         => ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A6,bNF_rel_fun(A,B,A,B,A6,A6)),times_times(A)),times_times(B)))
           => pp(aa(fun(list(B),B),bool,aa(fun(list(A),A),fun(fun(list(B),B),bool),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A6),A6),groups5270119922927024881d_list(A)),groups5270119922927024881d_list(B))) ) ) ) ).

% prod_list_transfer
tff(fact_6242_prod__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(list(A),A,groups5270119922927024881d_list(A),Xs)) ) ).

% prod_list.Cons
tff(fact_6243_prod__list_Oappend,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A),Ys: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),append(A,Xs,Ys)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups5270119922927024881d_list(A),Xs)),aa(list(A),A,groups5270119922927024881d_list(A),Ys)) ) ).

% prod_list.append
tff(fact_6244_division__segment__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),euclid7384307370059645450egment(A,A3)),aa(nat,A,semiring_1_of_nat(A),euclid6346220572633701492n_size(A,A3))) = A3 ) ).

% division_segment_euclidean_size
tff(fact_6245_division__segment__not__0,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A] : euclid7384307370059645450egment(A,A3) != zero_zero(A) ) ).

% division_segment_not_0
tff(fact_6246_division__segment__mult,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( euclid7384307370059645450egment(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),euclid7384307370059645450egment(A,A3)),euclid7384307370059645450egment(A,B2)) ) ) ) ) ).

% division_segment_mult
tff(fact_6247_prod__list__zero__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Xs: list(A)] :
          ( ( aa(list(A),A,groups5270119922927024881d_list(A),Xs) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% prod_list_zero_iff
tff(fact_6248_division__segment__mod,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( euclid7384307370059645450egment(A,modulo_modulo(A,A3,B2)) = euclid7384307370059645450egment(A,B2) ) ) ) ) ).

% division_segment_mod
tff(fact_6249_prod__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),Xs) = foldr(A,A,times_times(A),Xs,one_one(A)) ) ).

% prod_list.eq_foldr
tff(fact_6250_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B2: A] :
          ( ( euclid7384307370059645450egment(A,A3) = euclid7384307370059645450egment(A,B2) )
         => ( ( divide_divide(A,A3,B2) = zero_zero(A) )
          <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B2)))
              | ( B2 = zero_zero(A) ) ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
tff(fact_6251_div__bounded,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R: A,Q2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R)),euclid6346220572633701492n_size(A,B2)))
             => ( 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),Q2),B2)),R),B2) = Q2 ) ) ) ) ) ).

% div_bounded
tff(fact_6252_div__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R: A,Q2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R)),euclid6346220572633701492n_size(A,B2)))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B2)),R) = A3 )
               => ( divide_divide(A,A3,B2) = Q2 ) ) ) ) ) ) ).

% div_eqI
tff(fact_6253_mod__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R: A,Q2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),euclid6346220572633701492n_size(A,R)),euclid6346220572633701492n_size(A,B2)))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B2)),R) = A3 )
               => ( modulo_modulo(A,A3,B2) = R ) ) ) ) ) ) ).

% mod_eqI
tff(fact_6254_AboveS__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A6: set(A)] : order_AboveS(A,R,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_aef(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A6)) ).

% AboveS_def
tff(fact_6255_less__eq__enat__def,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),M2),N))
    <=> pp(extended_case_enat(bool,aTP_Lamp_aeg(extended_enat,fun(nat,bool),M2),fTrue,N)) ) ).

% less_eq_enat_def
tff(fact_6256_wo__rel_Osuc__greater,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B6: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),field2(A,R)))
       => ( ( order_AboveS(A,R,B6) != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),B6))
           => ( ( bNF_Wellorder_wo_suc(A,R,B6) != B2 )
              & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R,B6))),R)) ) ) ) ) ) ).

% wo_rel.suc_greater
tff(fact_6257_wo__rel_Oequals__suc__AboveS,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B6: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),order_AboveS(A,R,B6)))
         => ( ! [A14: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A14),order_AboveS(A,R,B6)))
               => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A14)),R)) )
           => ( A3 = bNF_Wellorder_wo_suc(A,R,B6) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
tff(fact_6258_wo__rel_Osuc__least__AboveS,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B6: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),order_AboveS(A,R,B6)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_Wellorder_wo_suc(A,R,B6)),A3)),R)) ) ) ).

% wo_rel.suc_least_AboveS
tff(fact_6259_wo__rel_Osuc__ofilter__in,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A6: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( order_ofilter(A,R,A6)
       => ( ( order_AboveS(A,R,A6) != bot_bot(set(A)) )
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R,A6))),R))
           => ( ( B2 != bNF_Wellorder_wo_suc(A,R,A6) )
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A6)) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
tff(fact_6260_sorted__wrt__iff__nth__Suc__transp,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),Xs: list(A)] :
      ( transp(A,P2)
     => ( sorted_wrt(A,P2,Xs)
      <=> ! [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,aa(A,fun(A,bool),P2,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I)))) ) ) ) ).

% sorted_wrt_iff_nth_Suc_transp
tff(fact_6261_transp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,ord_less_eq(A)) ) ).

% transp_le
tff(fact_6262_transp__ge,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_aeh(A,fun(A,bool))) ) ).

% transp_ge
tff(fact_6263_bsqr__max2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A12: A,A23: A,B1: A,B22: A] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( pp(aa(set(product_prod(product_prod(A,A),product_prod(A,A))),bool,aa(product_prod(product_prod(A,A),product_prod(A,A)),fun(set(product_prod(product_prod(A,A),product_prod(A,A))),bool),member(product_prod(product_prod(A,A),product_prod(A,A))),aa(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),fun(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A))),product_Pair(product_prod(A,A),product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),A23)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B1),B22))),bNF_Wellorder_bsqr(A,R)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,R,A12,A23)),bNF_We1388413361240627857o_max2(A,R,B1,B22))),R)) ) ) ).

% bsqr_max2
tff(fact_6264_bounded__bilinear__def,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
        <=> ( ! [A7: A,A15: A,B5: B] : aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),plus_plus(A),A7),A15)),B5) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A7),B5)),aa(B,C,aa(A,fun(B,C),Prod,A15),B5))
            & ! [A7: A,B5: B,B13: B] : aa(B,C,aa(A,fun(B,C),Prod,A7),aa(B,B,aa(B,fun(B,B),plus_plus(B),B5),B13)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A7),B5)),aa(B,C,aa(A,fun(B,C),Prod,A7),B13))
            & ! [R5: real,A7: A,B5: B] : aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),R5),A7)),B5) = aa(C,C,aa(real,fun(C,C),real_V8093663219630862766scaleR(C),R5),aa(B,C,aa(A,fun(B,C),Prod,A7),B5))
            & ! [A7: A,R5: real,B5: B] : aa(B,C,aa(A,fun(B,C),Prod,A7),aa(B,B,aa(real,fun(B,B),real_V8093663219630862766scaleR(B),R5),B5)) = aa(C,C,aa(real,fun(C,C),real_V8093663219630862766scaleR(C),R5),aa(B,C,aa(A,fun(B,C),Prod,A7),B5))
            & ? [K6: real] :
              ! [A7: A,B5: B] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A7),B5))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A7)),real_V7770717601297561774m_norm(B,B5))),K6))) ) ) ) ).

% bounded_bilinear_def
tff(fact_6265_bounded__bilinear_Ozero__left,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Prod: fun(A,fun(B,C)),B2: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,zero_zero(A)),B2) = zero_zero(C) ) ) ) ).

% bounded_bilinear.zero_left
tff(fact_6266_bounded__bilinear_Ozero__right,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Prod: fun(A,fun(B,C)),A3: A] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,A3),zero_zero(B)) = zero_zero(C) ) ) ) ).

% bounded_bilinear.zero_right
tff(fact_6267_bounded__bilinear__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => real_V2442710119149674383linear(A,A,A,times_times(A)) ) ).

% bounded_bilinear_mult
tff(fact_6268_well__order__on__domain,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_well_order_on(A,A6,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A6)) ) ) ) ).

% well_order_on_domain
tff(fact_6269_natLeq__Well__order,axiom,
    order_well_order_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Well_order
tff(fact_6270_bounded__bilinear_Obounded,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ? [K9: real] :
            ! [A8: A,B7: B] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A8),B7))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A8)),real_V7770717601297561774m_norm(B,B7))),K9))) ) ) ).

% bounded_bilinear.bounded
tff(fact_6271_natLeq__on__well__order__on,axiom,
    ! [N: nat] : order_well_order_on(nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_bt(nat,fun(nat,bool)),N)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aaw(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_well_order_on
tff(fact_6272_bounded__bilinear_Otendsto__right__zero,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C)),F3: fun(D,B),F4: filter(D),C3: A] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( filterlim(D,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_aei(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Prod),F3),C3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% bounded_bilinear.tendsto_right_zero
tff(fact_6273_bounded__bilinear_Otendsto__left__zero,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C)),F3: fun(D,A),F4: filter(D),C3: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( filterlim(D,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(D,C,aa(B,fun(D,C),aa(fun(D,A),fun(B,fun(D,C)),aTP_Lamp_aej(fun(A,fun(B,C)),fun(fun(D,A),fun(B,fun(D,C))),Prod),F3),C3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% bounded_bilinear.tendsto_left_zero
tff(fact_6274_bounded__bilinear_Otendsto__zero,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C)),F3: fun(D,A),F4: filter(D),G3: fun(D,B)] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( filterlim(D,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => ( filterlim(D,B,G3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
             => filterlim(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_aek(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),Prod),F3),G3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ) ).

% bounded_bilinear.tendsto_zero
tff(fact_6275_bounded__bilinear_Ononneg__bounded,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ? [K9: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K9))
              & ! [A8: A,B7: B] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A8),B7))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A8)),real_V7770717601297561774m_norm(B,B7))),K9))) ) ) ) ).

% bounded_bilinear.nonneg_bounded
tff(fact_6276_natLeq__on__Well__order,axiom,
    ! [N: nat] : order_well_order_on(nat,field2(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aaw(nat,fun(nat,fun(nat,bool)),N)))),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_aaw(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_Well_order
tff(fact_6277_Linear__order__Well__order__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => ( order_well_order_on(A,field2(A,R),R)
      <=> ! [A11: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A11),field2(A,R)))
           => ( ( A11 != bot_bot(set(A)) )
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A11))
                  & ! [Xa3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A11))
                     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R)) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_6278_bounded__bilinear_Opos__bounded,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ? [K9: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K9))
              & ! [A8: A,B7: B] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A8),B7))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A8)),real_V7770717601297561774m_norm(B,B7))),K9))) ) ) ) ).

% bounded_bilinear.pos_bounded
tff(fact_6279_bounded__bilinear_Ointro,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( ! [A5: A,A14: A,B4: B] : aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),plus_plus(A),A5),A14)),B4) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A5),B4)),aa(B,C,aa(A,fun(B,C),Prod,A14),B4))
         => ( ! [A5: A,B4: B,B10: B] : aa(B,C,aa(A,fun(B,C),Prod,A5),aa(B,B,aa(B,fun(B,B),plus_plus(B),B4),B10)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A5),B4)),aa(B,C,aa(A,fun(B,C),Prod,A5),B10))
           => ( ! [R3: real,A5: A,B4: B] : aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),R3),A5)),B4) = aa(C,C,aa(real,fun(C,C),real_V8093663219630862766scaleR(C),R3),aa(B,C,aa(A,fun(B,C),Prod,A5),B4))
             => ( ! [A5: A,R3: real,B4: B] : aa(B,C,aa(A,fun(B,C),Prod,A5),aa(B,B,aa(real,fun(B,B),real_V8093663219630862766scaleR(B),R3),B4)) = aa(C,C,aa(real,fun(C,C),real_V8093663219630862766scaleR(C),R3),aa(B,C,aa(A,fun(B,C),Prod,A5),B4))
               => ( ? [K8: real] :
                    ! [A5: A,B4: B] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A5),B4))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A5)),real_V7770717601297561774m_norm(B,B4))),K8)))
                 => real_V2442710119149674383linear(A,B,C,Prod) ) ) ) ) ) ) ).

% bounded_bilinear.intro
tff(fact_6280_mono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F3: fun(A,B),A6: fun(C,A),I5: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),image2(C,A,A6,I5)))
           => ( ( I5 != bot_bot(set(C)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ael(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A6),I5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),image2(C,A,A6,I5))))) ) ) ) ) ).

% mono_cSUP
tff(fact_6281_mono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),A6))
           => ( ( A6 != bot_bot(set(A)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image2(A,B,F3,A6))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),A6)))) ) ) ) ) ).

% mono_cSup
tff(fact_6282_bdd__above_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A),M5: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),M5)) )
         => pp(aa(set(A),bool,condit941137186595557371_above(A),A6)) ) ) ).

% bdd_above.I
tff(fact_6283_bdd__above_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,condit941137186595557371_above(A),A6))
        <=> ? [M9: A] :
            ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M9)) ) ) ) ).

% bdd_above.unfold
tff(fact_6284_bdd__above_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,condit941137186595557371_above(A),A6))
         => ~ ! [M8: A] :
                ~ ! [X5: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),M8)) ) ) ) ).

% bdd_above.E
tff(fact_6285_cSup__upper,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A,X6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),X6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ).

% cSup_upper
tff(fact_6286_cSup__upper2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A,X6: set(A),Y: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
           => ( pp(aa(set(A),bool,condit941137186595557371_above(A),X6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ) ).

% cSup_upper2
tff(fact_6287_bdd__above_OI2,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [A6: set(B),F3: fun(B,A),M5: A] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),M5)) )
         => pp(aa(set(A),bool,condit941137186595557371_above(A),image2(B,A,F3,A6))) ) ) ).

% bdd_above.I2
tff(fact_6288_cSUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A6: set(B),X: B,U: A] :
          ( pp(aa(set(A),bool,condit941137186595557371_above(A),image2(B,A,F3,A6)))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,X)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6)))) ) ) ) ) ).

% cSUP_upper2
tff(fact_6289_cSUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: B,A6: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A6))
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),image2(B,A,F3,A6)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X)),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6)))) ) ) ) ).

% cSUP_upper
tff(fact_6290_cSup__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B6: set(A),A6: set(A)] :
          ( ( B6 != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),A6))
           => ( ! [B4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B6))
                 => ? [X5: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A6))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B4),X5)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),B6)),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ) ) ).

% cSup_mono
tff(fact_6291_cSup__le__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S3: set(A),A3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),S3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S3)),A3))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),A3)) ) ) ) ) ) ).

% cSup_le_iff
tff(fact_6292_cSUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),F3: fun(B,A),U: A] :
          ( ( A6 != bot_bot(set(B)) )
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),image2(B,A,F3,A6)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),U))
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X4)),U)) ) ) ) ) ) ).

% cSUP_le_iff
tff(fact_6293_cSUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),G3: fun(C,A),B6: set(C),F3: fun(B,A)] :
          ( ( A6 != bot_bot(set(B)) )
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),image2(C,A,G3,B6)))
           => ( ! [N3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),N3),A6))
                 => ? [X5: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X5),B6))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,N3)),aa(C,A,G3,X5))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Sup_Sup(A),image2(C,A,G3,B6)))) ) ) ) ) ).

% cSUP_mono
tff(fact_6294_cSup__subset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(A),B6: set(A)] :
          ( ( A6 != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),B6))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B6))) ) ) ) ) ).

% cSup_subset_mono
tff(fact_6295_cSUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),G3: fun(B,A),B6: set(B),F3: fun(B,A)] :
          ( ( A6 != bot_bot(set(B)) )
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),image2(B,A,G3,B6)))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B6))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,G3,B6)))) ) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_6296_cSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(A),B6: set(A)] :
          ( pp(aa(set(A),bool,condit941137186595557371_above(A),A6))
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),B6))
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6) != bot_bot(set(A)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B6)))) ) ) ) ) ).

% cSup_inter_less_eq
tff(fact_6297_cSup__cInf,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S3: set(A)] :
          ( ( S3 != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),S3))
           => ( aa(set(A),A,complete_Sup_Sup(A),S3) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aem(set(A),fun(A,bool),S3))) ) ) ) ) ).

% cSup_cInf
tff(fact_6298_Max_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_max(A),aTP_Lamp_lw(A,fun(A,bool)),aTP_Lamp_aen(A,fun(A,bool))) ) ).

% Max.semilattice_order_set_axioms
tff(fact_6299_mono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),A6))
           => ( ( A6 != bot_bot(set(A)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A6))),aa(set(B),B,complete_Inf_Inf(B),image2(A,B,F3,A6)))) ) ) ) ) ).

% mono_cInf
tff(fact_6300_bdd__belowI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A),M2: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),X3)) )
         => pp(aa(set(A),bool,condit1013018076250108175_below(A),A6)) ) ) ).

% bdd_belowI
tff(fact_6301_bdd__below_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A),M5: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M5),X3)) )
         => pp(aa(set(A),bool,condit1013018076250108175_below(A),A6)) ) ) ).

% bdd_below.I
tff(fact_6302_bdd__below_OI2,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [A6: set(B),M5: A,F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M5),aa(B,A,F3,X3))) )
         => pp(aa(set(A),bool,condit1013018076250108175_below(A),image2(B,A,F3,A6))) ) ) ).

% bdd_below.I2
tff(fact_6303_bdd__belowI2,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [A6: set(B),M2: A,F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M2),aa(B,A,F3,X3))) )
         => pp(aa(set(A),bool,condit1013018076250108175_below(A),image2(B,A,F3,A6))) ) ) ).

% bdd_belowI2
tff(fact_6304_cInf__lower,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A,X6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),X6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),X)) ) ) ) ).

% cInf_lower
tff(fact_6305_cInf__lower2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X: A,X6: set(A),Y: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),X6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Y)) ) ) ) ) ).

% cInf_lower2
tff(fact_6306_bdd__below_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,condit1013018076250108175_below(A),A6))
         => ~ ! [M8: A] :
                ~ ! [X5: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M8),X5)) ) ) ) ).

% bdd_below.E
tff(fact_6307_bdd__below_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,condit1013018076250108175_below(A),A6))
        <=> ? [M9: A] :
            ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M9),X4)) ) ) ) ).

% bdd_below.unfold
tff(fact_6308_cINF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A6: set(B),X: B] :
          ( pp(aa(set(A),bool,condit1013018076250108175_below(A),image2(B,A,F3,A6)))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),aa(B,A,F3,X))) ) ) ) ).

% cINF_lower
tff(fact_6309_cINF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A6: set(B),X: B,U: A] :
          ( pp(aa(set(A),bool,condit1013018076250108175_below(A),image2(B,A,F3,A6)))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X)),U))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),U)) ) ) ) ) ).

% cINF_lower2
tff(fact_6310_cInf__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B6: set(A),A6: set(A)] :
          ( ( B6 != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),A6))
           => ( ! [B4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B6))
                 => ? [X5: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A6))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),B4)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B6))) ) ) ) ) ).

% cInf_mono
tff(fact_6311_le__cInf__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S3: set(A),A3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),S3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(set(A),A,complete_Inf_Inf(A),S3)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X4)) ) ) ) ) ) ).

% le_cInf_iff
tff(fact_6312_cINF__mono,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B6: set(B),F3: fun(C,A),A6: set(C),G3: fun(B,A)] :
          ( ( B6 != bot_bot(set(B)) )
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),image2(C,A,F3,A6)))
           => ( ! [M: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),M),B6))
                 => ? [X5: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X5),A6))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F3,X5)),aa(B,A,G3,M))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(C,A,F3,A6))),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,G3,B6)))) ) ) ) ) ).

% cINF_mono
tff(fact_6313_le__cINF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),F3: fun(B,A),U: A] :
          ( ( A6 != bot_bot(set(B)) )
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),image2(B,A,F3,A6)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))))
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,X4))) ) ) ) ) ) ).

% le_cINF_iff
tff(fact_6314_cInf__superset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(A),B6: set(A)] :
          ( ( A6 != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),B6))
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B6)),aa(set(A),A,complete_Inf_Inf(A),A6))) ) ) ) ) ).

% cInf_superset_mono
tff(fact_6315_Inf__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => lattic4895041142388067077er_set(A,inf_inf(A),ord_less_eq(A),ord_less(A)) ) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_6316_Min_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_min(A),ord_less_eq(A),ord_less(A)) ) ).

% Min.semilattice_order_set_axioms
tff(fact_6317_cINF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(B),G3: fun(B,A),B6: set(B),F3: fun(B,A)] :
          ( ( A6 != bot_bot(set(B)) )
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),image2(B,A,G3,B6)))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B6))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),B6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,G3,X3)),aa(B,A,F3,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,G3,B6))),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6)))) ) ) ) ) ) ).

% cINF_superset_mono
tff(fact_6318_less__eq__cInf__inter,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(A),B6: set(A)] :
          ( pp(aa(set(A),bool,condit1013018076250108175_below(A),A6))
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),B6))
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6) != bot_bot(set(A)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6)))) ) ) ) ) ).

% less_eq_cInf_inter
tff(fact_6319_cInf__le__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A6: set(A)] :
          ( ( A6 != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,condit941137186595557371_above(A),A6))
           => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),A6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ) ) ).

% cInf_le_cSup
tff(fact_6320_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => lattic4895041142388067077er_set(A,sup_sup(A),aTP_Lamp_aeo(A,fun(A,bool)),aTP_Lamp_aep(A,fun(A,bool))) ) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_6321_cInf__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S3: set(A)] :
          ( ( S3 != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),S3))
           => ( aa(set(A),A,complete_Inf_Inf(A),S3) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aeq(set(A),fun(A,bool),S3))) ) ) ) ) ).

% cInf_cSup
tff(fact_6322_mono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(A) )
     => ! [F3: fun(A,B),A6: fun(C,A),I5: set(C)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),image2(C,A,A6,I5)))
           => ( ( I5 != bot_bot(set(C)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),image2(C,A,A6,I5)))),aa(set(B),B,complete_Inf_Inf(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ael(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A6),I5)))) ) ) ) ) ).

% mono_cINF
tff(fact_6323_Gcd__fin__def,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Gcd_fin(A) = bounde2362111253966948842tice_F(A,gcd_gcd(A),zero_zero(A),one_one(A)) ) ) ).

% Gcd_fin_def
tff(fact_6324_flip__pred,axiom,
    ! [A: $tType,B: $tType,A6: set(product_prod(A,B)),R2: fun(B,fun(A,bool))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),conversep(B,A,R2)))))
     => pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),image2(product_prod(A,B),product_prod(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_xi(A,fun(B,product_prod(B,A)))),A6)),aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),R2)))) ) ).

% flip_pred
tff(fact_6325_gen__length__def,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,gen_length(A,N),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)) ).

% gen_length_def
tff(fact_6326_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,M2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),N))
         => ( aa(A,char,unique5772411509450598832har_of(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),M2)) = aa(A,char,unique5772411509450598832har_of(A),M2) ) ) ) ).

% char_of_take_bit_eq
tff(fact_6327_inj__on__char__of__nat,axiom,
    inj_on(nat,char,unique5772411509450598832har_of(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% inj_on_char_of_nat
tff(fact_6328_UNIV__char__of__nat,axiom,
    top_top(set(char)) = image2(nat,char,unique5772411509450598832har_of(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_6329_gen__length__code_I2_J,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] : aa(list(B),nat,gen_length(B,N),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(B),nat,gen_length(B,aa(nat,nat,suc,N)),Xs) ).

% gen_length_code(2)
tff(fact_6330_length__code,axiom,
    ! [A: $tType] : size_size(list(A)) = gen_length(A,zero_zero(nat)) ).

% length_code
tff(fact_6331_range__nat__of__char,axiom,
    image2(char,nat,comm_s6883823935334413003f_char(nat),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% range_nat_of_char
tff(fact_6332_relImage__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,B)),F3: fun(B,A)] : bNF_Gr4221423524335903396lImage(B,A,R2,F3) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),bool),aTP_Lamp_aer(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),R2),F3)) ).

% relImage_def
tff(fact_6333_card__def,axiom,
    ! [B: $tType] : finite_card(B) = finite_folding_F(B,nat,aTP_Lamp_aes(B,fun(nat,nat)),zero_zero(nat)) ).

% card_def
tff(fact_6334_mono__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType] :
      ( ( order(B)
        & order(D)
        & order(C)
        & order(A) )
     => ! [A6: fun(A,fun(B,bool)),B6: fun(C,fun(D,bool))] :
          ( bi_total(A,B,A6)
         => ( pp(aa(fun(B,fun(B,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(B,fun(B,bool)),bool),bNF_rel_fun(A,B,fun(A,bool),fun(B,bool),A6,bNF_rel_fun(A,B,bool,bool,A6,fequal(bool))),ord_less_eq(A)),ord_less_eq(B)))
           => ( pp(aa(fun(D,fun(D,bool)),bool,aa(fun(C,fun(C,bool)),fun(fun(D,fun(D,bool)),bool),bNF_rel_fun(C,D,fun(C,bool),fun(D,bool),B6,bNF_rel_fun(C,D,bool,bool,B6,fequal(bool))),ord_less_eq(C)),ord_less_eq(D)))
             => pp(aa(fun(fun(B,D),bool),bool,aa(fun(fun(A,C),bool),fun(fun(fun(B,D),bool),bool),bNF_rel_fun(fun(A,C),fun(B,D),bool,bool,bNF_rel_fun(A,B,C,D,A6,B6),fequal(bool)),order_mono(A,C)),order_mono(B,D))) ) ) ) ) ).

% mono_transfer
tff(fact_6335_plus__rat__def,axiom,
    plus_plus(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_aed(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat_def
tff(fact_6336_diff__rat,axiom,
    ! [B2: int,D3: int,A3: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),fract(A3,B2)),fract(C3,D3)) = fract(aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)) ) ) ) ).

% diff_rat
tff(fact_6337_sgn__rat,axiom,
    ! [A3: int,B2: int] : aa(rat,rat,sgn_sgn(rat),fract(A3,B2)) = aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),A3)),aa(int,int,sgn_sgn(int),B2))) ).

% sgn_rat
tff(fact_6338_mult__rat,axiom,
    ! [A3: int,B2: int,C3: int,D3: int] : aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),fract(A3,B2)),fract(C3,D3)) = fract(aa(int,int,aa(int,fun(int,int),times_times(int),A3),C3),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)) ).

% mult_rat
tff(fact_6339_divide__rat,axiom,
    ! [A3: int,B2: int,C3: int,D3: int] : divide_divide(rat,fract(A3,B2),fract(C3,D3)) = fract(aa(int,int,aa(int,fun(int,int),times_times(int),A3),D3),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C3)) ).

% divide_rat
tff(fact_6340_less__rat,axiom,
    ! [B2: int,D3: int,A3: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),fract(A3,B2)),fract(C3,D3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)))) ) ) ) ).

% less_rat
tff(fact_6341_add__rat,axiom,
    ! [B2: int,D3: int,A3: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),fract(A3,B2)),fract(C3,D3)) = fract(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)) ) ) ) ).

% add_rat
tff(fact_6342_le__rat,axiom,
    ! [B2: int,D3: int,A3: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),fract(A3,B2)),fract(C3,D3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)))) ) ) ) ).

% le_rat
tff(fact_6343_mult__rat__cancel,axiom,
    ! [C3: int,A3: int,B2: int] :
      ( ( C3 != zero_zero(int) )
     => ( fract(aa(int,int,aa(int,fun(int,int),times_times(int),C3),A3),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)) = fract(A3,B2) ) ) ).

% mult_rat_cancel
tff(fact_6344_eq__rat_I1_J,axiom,
    ! [B2: int,D3: int,A3: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( ( fract(A3,B2) = fract(C3,D3) )
        <=> ( aa(int,int,aa(int,fun(int,int),times_times(int),A3),D3) = aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2) ) ) ) ) ).

% eq_rat(1)
tff(fact_6345_positive__rat,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(rat,bool,positive,fract(A3,B2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A3),B2))) ) ).

% positive_rat
tff(fact_6346_times__rat__def,axiom,
    times_times(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_aec(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat_def
tff(fact_6347_irrefl__tranclI,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A] :
      ( ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),converse(A,A,R)),transitive_rtrancl(A,R)) = bot_bot(set(product_prod(A,A))) )
     => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),transitive_trancl(A,R))) ) ).

% irrefl_tranclI
tff(fact_6348_prod__set__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : basic_snds(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = aa(set(B),set(B),insert(B,Y),bot_bot(set(B))) ).

% prod_set_simps(2)
tff(fact_6349_converse__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),converse(B,A,R)))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),R)) ) ).

% converse_iff
tff(fact_6350_conversep__converse__eq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),X5: B,Xa: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X5),Xa))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X5),Xa)),converse(A,B,R))) ) ).

% conversep_converse_eq
tff(fact_6351_converse__unfold,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A))] : converse(B,A,R) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_aet(set(product_prod(B,A)),fun(A,fun(B,bool)),R))) ).

% converse_unfold
tff(fact_6352_trancl__converseI,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),converse(A,A,transitive_trancl(A,R))))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,converse(A,A,R)))) ) ).

% trancl_converseI
tff(fact_6353_trancl__converseD,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,converse(A,A,R))))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),converse(A,A,transitive_trancl(A,R)))) ) ).

% trancl_converseD
tff(fact_6354_converseI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
     => pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),converse(A,B,R))) ) ).

% converseI
tff(fact_6355_converseE,axiom,
    ! [A: $tType,B: $tType,Yx: product_prod(A,B),R: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),Yx),converse(B,A,R)))
     => ~ ! [X3: B,Y3: A] :
            ( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y3),X3) )
           => ~ pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),Y3)),R)) ) ) ).

% converseE
tff(fact_6356_converseD,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),converse(B,A,R)))
     => pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),R)) ) ).

% converseD
tff(fact_6357_converse_Osimps,axiom,
    ! [B: $tType,A: $tType,A12: B,A23: A,R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A12),A23)),converse(A,B,R)))
    <=> ? [A7: A,B5: B] :
          ( ( A12 = B5 )
          & ( A23 = A7 )
          & pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5)),R)) ) ) ).

% converse.simps
tff(fact_6358_converse_Ocases,axiom,
    ! [B: $tType,A: $tType,A12: B,A23: A,R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A12),A23)),converse(A,B,R)))
     => pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A23),A12)),R)) ) ).

% converse.cases
tff(fact_6359_rtrancl__converseI,axiom,
    ! [A: $tType,Y: A,X: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),transitive_rtrancl(A,R)))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,converse(A,A,R)))) ) ).

% rtrancl_converseI
tff(fact_6360_rtrancl__converseD,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,converse(A,A,R))))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),transitive_rtrancl(A,R))) ) ).

% rtrancl_converseD
tff(fact_6361_converse__def,axiom,
    ! [B: $tType,A: $tType,X5: set(product_prod(A,B))] : converse(A,B,X5) = aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),X5)))) ).

% converse_def
tff(fact_6362_prod__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B] : basic_fsts(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ).

% prod_set_simps(1)
tff(fact_6363_integer__of__char__code,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : integer_of_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(bool,code_integer,zero_neq_one_of_bool(code_integer),B72)),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B62))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B52))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B42))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B32))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B22))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B1))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B0)) ).

% integer_of_char_code
tff(fact_6364_char_Osize_I2_J,axiom,
    ! [X1: bool,X2: bool,X33: bool,X42: bool,X52: bool,X62: bool,X72: bool,X8: bool] : aa(char,nat,size_size(char),char2(X1,X2,X33,X42,X52,X62,X72,X8)) = zero_zero(nat) ).

% char.size(2)
tff(fact_6365_char_Osize__gen,axiom,
    ! [X1: bool,X2: bool,X33: bool,X42: bool,X52: bool,X62: bool,X72: bool,X8: bool] : size_char(char2(X1,X2,X33,X42,X52,X62,X72,X8)) = zero_zero(nat) ).

% char.size_gen
tff(fact_6366_numeral__le__enat__iff,axiom,
    ! [M2: num,N: nat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M2)),extended_enat2(N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),M2)),N)) ) ).

% numeral_le_enat_iff
tff(fact_6367_idiff__enat__0__right,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),N),extended_enat2(zero_zero(nat))) = N ).

% idiff_enat_0_right
tff(fact_6368_idiff__enat__0,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),extended_enat2(zero_zero(nat))),N) = extended_enat2(zero_zero(nat)) ).

% idiff_enat_0
tff(fact_6369_times__enat__simps_I1_J,axiom,
    ! [M2: nat,N: nat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extended_enat2(M2)),extended_enat2(N)) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N)) ).

% times_enat_simps(1)
tff(fact_6370_enat__ord__simps_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),extended_enat2(M2)),extended_enat2(N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N)) ) ).

% enat_ord_simps(1)
tff(fact_6371_Suc__ile__eq,axiom,
    ! [M2: nat,N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),extended_enat2(aa(nat,nat,suc,M2))),N))
    <=> pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),extended_enat2(M2)),N)) ) ).

% Suc_ile_eq
tff(fact_6372_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_zero(extended_enat) = extended_enat2(X) )
    <=> ( X = zero_zero(nat) ) ) ).

% enat_0_iff(2)
tff(fact_6373_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( extended_enat2(X) = zero_zero(extended_enat) )
    <=> ( X = zero_zero(nat) ) ) ).

% enat_0_iff(1)
tff(fact_6374_zero__enat__def,axiom,
    zero_zero(extended_enat) = extended_enat2(zero_zero(nat)) ).

% zero_enat_def
tff(fact_6375_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,N: nat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),X),Y)),extended_enat2(N)))
    <=> ? [Y8: nat,X9: nat] :
          ( ( X = extended_enat2(X9) )
          & ( Y = extended_enat2(Y8) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X9),Y8)),N)) ) ) ).

% iadd_le_enat_iff
tff(fact_6376_elimnum,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary),N)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) ) ).

% elimnum
tff(fact_6377_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,L: nat] : vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extended_enat2(L)) = vEBT_Node(Info2,Deg,take(vEBT_VEBT,divide_divide(nat,L,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_aeu(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg),TreeList2)),vEBT_VEBT_elim_dead(Summary,extended_enat2(divide_divide(nat,L,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ).

% VEBT_internal.elim_dead.simps(3)
tff(fact_6378_enat__0__less__mult__iff,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M2),N)))
    <=> ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),M2))
        & pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),N)) ) ) ).

% enat_0_less_mult_iff
tff(fact_6379_imult__is__0,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M2),N) = zero_zero(extended_enat) )
    <=> ( ( M2 = zero_zero(extended_enat) )
        | ( N = zero_zero(extended_enat) ) ) ) ).

% imult_is_0
tff(fact_6380_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A3: bool,B2: bool,Uu: extended_enat] : vEBT_VEBT_elim_dead(vEBT_Leaf(A3,B2),Uu) = vEBT_Leaf(A3,B2) ).

% VEBT_internal.elim_dead.simps(1)
tff(fact_6381_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X,Xa2) = Y )
     => ( ! [A5: bool,B4: bool] :
            ( ( X = vEBT_Leaf(A5,B4) )
           => ( Y != vEBT_Leaf(A5,B4) ) )
       => ( ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
             => ( ( Xa2 = extend4730790105801354508finity(extended_enat) )
               => ( Y != vEBT_Node(Info,Deg2,map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_aeu(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg2),TreeList),vEBT_VEBT_elim_dead(Summary2,extend4730790105801354508finity(extended_enat))) ) ) )
         => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
               => ! [L3: nat] :
                    ( ( Xa2 = extended_enat2(L3) )
                   => ( Y != vEBT_Node(Info,Deg2,take(vEBT_VEBT,divide_divide(nat,L3,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))))),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_aeu(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg2),TreeList)),vEBT_VEBT_elim_dead(Summary2,extended_enat2(divide_divide(nat,L3,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_6382_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extend4730790105801354508finity(extended_enat)) = vEBT_Node(Info2,Deg,map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_aeu(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg),TreeList2),vEBT_VEBT_elim_dead(Summary,extend4730790105801354508finity(extended_enat))) ).

% VEBT_internal.elim_dead.simps(2)
tff(fact_6383_elimcomplete,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info2,Deg,TreeList2,Summary),N)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info2,Deg,TreeList2,Summary),extend4730790105801354508finity(extended_enat)) = vEBT_Node(Info2,Deg,TreeList2,Summary) ) ) ).

% elimcomplete
tff(fact_6384_times__enat__simps_I2_J,axiom,
    aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extend4730790105801354508finity(extended_enat)),extend4730790105801354508finity(extended_enat)) = extend4730790105801354508finity(extended_enat) ).

% times_enat_simps(2)
tff(fact_6385_times__enat__simps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(N)) = zero_zero(extended_enat) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(N)) = extend4730790105801354508finity(extended_enat) ) ) ) ).

% times_enat_simps(3)
tff(fact_6386_times__enat__simps_I4_J,axiom,
    ! [M2: nat] :
      ( ( ( M2 = zero_zero(nat) )
       => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extended_enat2(M2)),extend4730790105801354508finity(extended_enat)) = zero_zero(extended_enat) ) )
      & ( ( M2 != zero_zero(nat) )
       => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extended_enat2(M2)),extend4730790105801354508finity(extended_enat)) = extend4730790105801354508finity(extended_enat) ) ) ) ).

% times_enat_simps(4)
tff(fact_6387_imult__is__infinity,axiom,
    ! [A3: extended_enat,B2: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),A3),B2) = extend4730790105801354508finity(extended_enat) )
    <=> ( ( ( A3 = extend4730790105801354508finity(extended_enat) )
          & ( B2 != zero_zero(extended_enat) ) )
        | ( ( B2 = extend4730790105801354508finity(extended_enat) )
          & ( A3 != zero_zero(extended_enat) ) ) ) ) ).

% imult_is_infinity
tff(fact_6388_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,extended_enat)] :
      ( ! [A5: bool,B4: bool,Uu2: extended_enat] : X != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),vEBT_Leaf(A5,B4)),Uu2)
     => ( ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] : X != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),vEBT_Node(Info,Deg2,TreeList,Summary2)),extend4730790105801354508finity(extended_enat))
       => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,L3: nat] : X != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),vEBT_Node(Info,Deg2,TreeList,Summary2)),extended_enat2(L3)) ) ) ).

% VEBT_internal.elim_dead.cases
tff(fact_6389_imult__infinity,axiom,
    ! [N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),N))
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extend4730790105801354508finity(extended_enat)),N) = extend4730790105801354508finity(extended_enat) ) ) ).

% imult_infinity
tff(fact_6390_imult__infinity__right,axiom,
    ! [N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),N))
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),N),extend4730790105801354508finity(extended_enat)) = extend4730790105801354508finity(extended_enat) ) ) ).

% imult_infinity_right
tff(fact_6391_times__enat__def,axiom,
    ! [M2: extended_enat,N: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M2),N) = extended_case_enat(extended_enat,aTP_Lamp_aew(extended_enat,fun(nat,extended_enat),N),if(extended_enat,aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),fequal(extended_enat),N),zero_zero(extended_enat)),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)),M2) ).

% times_enat_def
tff(fact_6392_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X,Xa2) = Y )
     => ( accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),X),Xa2))
       => ( ! [A5: bool,B4: bool] :
              ( ( X = vEBT_Leaf(A5,B4) )
             => ( ( Y = vEBT_Leaf(A5,B4) )
               => ~ accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),vEBT_Leaf(A5,B4)),Xa2)) ) )
         => ( ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
               => ( ( Xa2 = extend4730790105801354508finity(extended_enat) )
                 => ( ( Y = vEBT_Node(Info,Deg2,map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_aeu(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg2),TreeList),vEBT_VEBT_elim_dead(Summary2,extend4730790105801354508finity(extended_enat))) )
                   => ~ accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),vEBT_Node(Info,Deg2,TreeList,Summary2)),extend4730790105801354508finity(extended_enat))) ) ) )
           => ~ ! [Info: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Info,Deg2,TreeList,Summary2) )
                 => ! [L3: nat] :
                      ( ( Xa2 = extended_enat2(L3) )
                     => ( ( Y = vEBT_Node(Info,Deg2,take(vEBT_VEBT,divide_divide(nat,L3,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))))),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_aeu(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg2),TreeList)),vEBT_VEBT_elim_dead(Summary2,extended_enat2(divide_divide(nat,L3,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) )
                       => ~ accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel,aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),aa(vEBT_VEBT,fun(extended_enat,product_prod(vEBT_VEBT,extended_enat)),product_Pair(vEBT_VEBT,extended_enat),vEBT_Node(Info,Deg2,TreeList,Summary2)),extended_enat2(L3))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_6393_eSuc__def,axiom,
    ! [I2: extended_enat] : extended_eSuc(I2) = extended_case_enat(extended_enat,aTP_Lamp_aex(nat,extended_enat),extend4730790105801354508finity(extended_enat),I2) ).

% eSuc_def
tff(fact_6394_binomial__def,axiom,
    ! [N: nat,K2: nat] : aa(nat,nat,binomial(N),K2) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),bool),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),bool),aTP_Lamp_aey(nat,fun(nat,fun(set(nat),bool)),N),K2))) ).

% binomial_def
tff(fact_6395_enat__eSuc__iff,axiom,
    ! [Y: nat,X: extended_enat] :
      ( ( extended_enat2(Y) = extended_eSuc(X) )
    <=> ? [N5: nat] :
          ( ( Y = aa(nat,nat,suc,N5) )
          & ( extended_enat2(N5) = X ) ) ) ).

% enat_eSuc_iff
tff(fact_6396_eSuc__enat__iff,axiom,
    ! [X: extended_enat,Y: nat] :
      ( ( extended_eSuc(X) = extended_enat2(Y) )
    <=> ? [N5: nat] :
          ( ( Y = aa(nat,nat,suc,N5) )
          & ( X = extended_enat2(N5) ) ) ) ).

% eSuc_enat_iff
tff(fact_6397_eSuc__enat,axiom,
    ! [N: nat] : extended_eSuc(extended_enat2(N)) = extended_enat2(aa(nat,nat,suc,N)) ).

% eSuc_enat
tff(fact_6398_mult__eSuc__right,axiom,
    ! [M2: extended_enat,N: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M2),extended_eSuc(N)) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),M2),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M2),N)) ).

% mult_eSuc_right
tff(fact_6399_mult__eSuc,axiom,
    ! [M2: extended_enat,N: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extended_eSuc(M2)),N) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),N),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M2),N)) ).

% mult_eSuc
tff(fact_6400_Nats__altdef2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( semiring_1_Nats(A) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aez(A,bool)) ) ) ).

% Nats_altdef2
tff(fact_6401_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y)),less_than))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y)) ) ).

% less_than_iff
tff(fact_6402_of__nat__in__Nats,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_1_Nats(A))) ) ).

% of_nat_in_Nats
tff(fact_6403_Nats__induct,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: A,P2: fun(A,bool)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),semiring_1_Nats(A)))
         => ( ! [N3: nat] : pp(aa(A,bool,P2,aa(nat,A,semiring_1_of_nat(A),N3)))
           => pp(aa(A,bool,P2,X)) ) ) ) ).

% Nats_induct
tff(fact_6404_Nats__cases,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),semiring_1_Nats(A)))
         => ~ ! [N3: nat] : X != aa(nat,A,semiring_1_of_nat(A),N3) ) ) ).

% Nats_cases
tff(fact_6405_Nats__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),one_one(A)),semiring_1_Nats(A))) ) ).

% Nats_1
tff(fact_6406_Nats__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),semiring_1_Nats(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),semiring_1_Nats(A))) ) ) ) ).

% Nats_mult
tff(fact_6407_Nats__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),semiring_1_Nats(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),semiring_1_Nats(A))) ) ) ) ).

% Nats_add
tff(fact_6408_Nats__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),semiring_1_Nats(A))) ) ).

% Nats_0
tff(fact_6409_Nats__diff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),semiring_1_Nats(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),semiring_1_Nats(A))) ) ) ) ) ).

% Nats_diff
tff(fact_6410_Nats__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( semiring_1_Nats(A) = image2(nat,A,semiring_1_of_nat(A),top_top(set(nat))) ) ) ).

% Nats_def
tff(fact_6411_lenlex__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lenlex(A,R) = inv_image(product_prod(nat,list(A)),list(A),lex_prod(nat,list(A),less_than,lex(A,R)),aTP_Lamp_afa(list(A),product_prod(nat,list(A)))) ).

% lenlex_def
tff(fact_6412_mlex__prod__def,axiom,
    ! [A: $tType,F3: fun(A,nat),R2: set(product_prod(A,A))] : mlex_prod(A,F3,R2) = inv_image(product_prod(nat,A),A,lex_prod(nat,A,less_than,R2),aTP_Lamp_afb(fun(A,nat),fun(A,product_prod(nat,A)),F3)) ).

% mlex_prod_def
tff(fact_6413_in__inv__image,axiom,
    ! [A: $tType,B: $tType,X: A,Y: A,R: set(product_prod(B,B)),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),inv_image(B,A,R,F3)))
    <=> pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F3,X)),aa(A,B,F3,Y))),R)) ) ).

% in_inv_image
tff(fact_6414_inv__image__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(B,B)),F3: fun(A,B)] : inv_image(B,A,R,F3) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_afc(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,bool))),R),F3))) ).

% inv_image_def
tff(fact_6415_Real_Opositive_Orsp,axiom,
    pp(aa(fun(fun(nat,rat),bool),bool,aa(fun(fun(nat,rat),bool),fun(fun(fun(nat,rat),bool),bool),bNF_rel_fun(fun(nat,rat),fun(nat,rat),bool,bool,realrel,fequal(bool)),aTP_Lamp_afd(fun(nat,rat),bool)),aTP_Lamp_afd(fun(nat,rat),bool))) ).

% Real.positive.rsp
tff(fact_6416_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list(nat)] : gcd_Gcd(nat,aa(list(nat),set(nat),set2(nat),Xs)) = fold(nat,nat,gcd_gcd(nat),Xs,zero_zero(nat)) ).

% Gcd_nat_set_eq_fold
tff(fact_6417_times__real_Orsp,axiom,
    pp(aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),bool,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),bool),bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),fun(fun(nat,rat),fun(nat,rat)),realrel,bNF_rel_fun(fun(nat,rat),fun(nat,rat),fun(nat,rat),fun(nat,rat),realrel,realrel)),aTP_Lamp_afe(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),aTP_Lamp_afe(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))))) ).

% times_real.rsp
tff(fact_6418_Gcd__set__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xs: list(A)] : gcd_Gcd(A,aa(list(A),set(A),set2(A),Xs)) = fold(A,A,gcd_gcd(A),Xs,zero_zero(A)) ) ).

% Gcd_set_eq_fold
tff(fact_6419_Gcd__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(list(A),set(A),set2(A),Xs)) = fold(A,A,gcd_gcd(A),Xs,zero_zero(A)) ) ).

% Gcd_fin.set_eq_fold
tff(fact_6420_of__rat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),field_char_0_of_rat(A,R)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R),zero_zero(rat))) ) ) ).

% of_rat_le_0_iff
tff(fact_6421_zero__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),field_char_0_of_rat(A,R)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),R)) ) ) ).

% zero_le_of_rat_iff
tff(fact_6422_zero__eq__of__rat__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( zero_zero(A) = field_char_0_of_rat(A,A3) )
        <=> ( zero_zero(rat) = A3 ) ) ) ).

% zero_eq_of_rat_iff
tff(fact_6423_of__rat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( field_char_0_of_rat(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(rat) ) ) ) ).

% of_rat_eq_0_iff
tff(fact_6424_of__rat__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( field_char_0_of_rat(A,zero_zero(rat)) = zero_zero(A) ) ) ).

% of_rat_0
tff(fact_6425_of__rat__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),field_char_0_of_rat(A,R)),one_one(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R),one_one(rat))) ) ) ).

% of_rat_le_1_iff
tff(fact_6426_one__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),field_char_0_of_rat(A,R)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),R)) ) ) ).

% one_le_of_rat_iff
tff(fact_6427_zero__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),field_char_0_of_rat(A,R)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R)) ) ) ).

% zero_less_of_rat_iff
tff(fact_6428_of__rat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),field_char_0_of_rat(A,R)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R),zero_zero(rat))) ) ) ).

% of_rat_less_0_iff
tff(fact_6429_of__rat__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat,B2: rat] : field_char_0_of_rat(A,aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),field_char_0_of_rat(A,A3)),field_char_0_of_rat(A,B2)) ) ).

% of_rat_mult
tff(fact_6430_of__rat__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat,S2: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),field_char_0_of_rat(A,R)),field_char_0_of_rat(A,S2)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R),S2)) ) ) ).

% of_rat_less_eq
tff(fact_6431_num_Orec__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,bool))] : pp(aa(fun(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)))),bool,aa(fun(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)))),fun(fun(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)))),bool),bNF_rel_fun(A,B,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B))),S3,bNF_rel_fun(fun(num,fun(A,A)),fun(num,fun(B,B)),fun(fun(num,fun(A,A)),fun(num,A)),fun(fun(num,fun(B,B)),fun(num,B)),bNF_rel_fun(num,num,fun(A,A),fun(B,B),fequal(num),bNF_rel_fun(A,B,A,B,S3,S3)),bNF_rel_fun(fun(num,fun(A,A)),fun(num,fun(B,B)),fun(num,A),fun(num,B),bNF_rel_fun(num,num,fun(A,A),fun(B,B),fequal(num),bNF_rel_fun(A,B,A,B,S3,S3)),bNF_rel_fun(num,num,A,B,fequal(num),S3)))),rec_num(A)),rec_num(B))) ).

% num.rec_transfer
tff(fact_6432_Real_Opositive_Otransfer,axiom,
    pp(aa(fun(real,bool),bool,aa(fun(fun(nat,rat),bool),fun(fun(real,bool),bool),bNF_rel_fun(fun(nat,rat),real,bool,bool,pcr_real,fequal(bool)),aTP_Lamp_afd(fun(nat,rat),bool)),positive2)) ).

% Real.positive.transfer
tff(fact_6433_Real_Opositive__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,positive2,X))
     => ( pp(aa(real,bool,positive2,Y))
       => pp(aa(real,bool,positive2,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y))) ) ) ).

% Real.positive_mult
tff(fact_6434_times__real_Otransfer,axiom,
    pp(aa(fun(real,fun(real,real)),bool,aa(fun(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),fun(fun(real,fun(real,real)),bool),bNF_rel_fun(fun(nat,rat),real,fun(fun(nat,rat),fun(nat,rat)),fun(real,real),pcr_real,bNF_rel_fun(fun(nat,rat),real,fun(nat,rat),real,pcr_real,pcr_real)),aTP_Lamp_afe(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)))),times_times(real))) ).

% times_real.transfer
tff(fact_6435_num_Ocase__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,bool))] : pp(aa(fun(B,fun(fun(num,B),fun(fun(num,B),fun(num,B)))),bool,aa(fun(A,fun(fun(num,A),fun(fun(num,A),fun(num,A)))),fun(fun(B,fun(fun(num,B),fun(fun(num,B),fun(num,B)))),bool),bNF_rel_fun(A,B,fun(fun(num,A),fun(fun(num,A),fun(num,A))),fun(fun(num,B),fun(fun(num,B),fun(num,B))),S3,bNF_rel_fun(fun(num,A),fun(num,B),fun(fun(num,A),fun(num,A)),fun(fun(num,B),fun(num,B)),bNF_rel_fun(num,num,A,B,fequal(num),S3),bNF_rel_fun(fun(num,A),fun(num,B),fun(num,A),fun(num,B),bNF_rel_fun(num,num,A,B,fequal(num),S3),bNF_rel_fun(num,num,A,B,fequal(num),S3)))),case_num(A)),case_num(B))) ).

% num.case_transfer
tff(fact_6436_Real_Opositive_Orep__eq,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,positive2,X))
    <=> ? [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
          & ? [K3: nat] :
            ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,aa(real,fun(nat,rat),rep_real,X),N5))) ) ) ) ).

% Real.positive.rep_eq
tff(fact_6437_num_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: A,F22: fun(num,A),F32: fun(num,A),Num: num] : aa(A,B,H,aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),F1),F22),F32),Num)) = aa(num,B,aa(fun(num,B),fun(num,B),aa(fun(num,B),fun(fun(num,B),fun(num,B)),aa(B,fun(fun(num,B),fun(fun(num,B),fun(num,B))),case_num(B),aa(A,B,H,F1)),aa(fun(num,A),fun(num,B),aTP_Lamp_aff(fun(A,B),fun(fun(num,A),fun(num,B)),H),F22)),aa(fun(num,A),fun(num,B),aTP_Lamp_aff(fun(A,B),fun(fun(num,A),fun(num,B)),H),F32)),Num) ).

% num.case_distrib
tff(fact_6438_Real_Opositive__def,axiom,
    positive2 = aa(fun(fun(nat,rat),bool),fun(real,bool),map_fun(real,fun(nat,rat),bool,bool,rep_real,id(bool)),aTP_Lamp_afd(fun(nat,rat),bool)) ).

% Real.positive_def
tff(fact_6439_Real_Opositive_Oabs__eq,axiom,
    ! [X: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X),X))
     => ( pp(aa(real,bool,positive2,real2(X)))
      <=> ? [R5: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
            & ? [K3: nat] :
              ! [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,X,N5))) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_6440_times__real_Oabs__eq,axiom,
    ! [Xa2: fun(nat,rat),X: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,Xa2),Xa2))
     => ( pp(aa(fun(nat,rat),bool,aa(fun(nat,rat),fun(fun(nat,rat),bool),realrel,X),X))
       => ( aa(real,real,aa(real,fun(real,real),times_times(real),real2(Xa2)),real2(X)) = real2(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_afe(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Xa2),X)) ) ) ) ).

% times_real.abs_eq
tff(fact_6441_le__Real,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real2(X6)),real2(Y6)))
        <=> ! [R5: rat] :
              ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
             => ? [K3: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
                 => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(nat,rat,X6,N5)),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(nat,rat,Y6,N5)),R5))) ) ) ) ) ) ).

% le_Real
tff(fact_6442_not__positive__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ pp(aa(real,bool,positive2,real2(X6)))
      <=> ! [R5: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
           => ? [K3: nat] :
              ! [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(nat,rat,X6,N5)),R5)) ) ) ) ) ).

% not_positive_Real
tff(fact_6443_cauchy__mult,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => cauchy(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_afe(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ).

% cauchy_mult
tff(fact_6444_mult__Real,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( cauchy(Y6)
       => ( aa(real,real,aa(real,fun(real,real),times_times(real),real2(X6)),real2(Y6)) = real2(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_afe(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ) ).

% mult_Real
tff(fact_6445_cauchyD,axiom,
    ! [X6: fun(nat,rat),R: rat] :
      ( cauchy(X6)
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R))
       => ? [K: nat] :
          ! [M3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M3))
           => ! [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N4))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M3)),aa(nat,rat,X6,N4)))),R)) ) ) ) ) ).

% cauchyD
tff(fact_6446_cauchyI,axiom,
    ! [X6: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R3))
         => ? [K4: nat] :
            ! [M: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K4),M))
             => ! [N3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K4),N3))
                 => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M)),aa(nat,rat,X6,N3)))),R3)) ) ) )
     => cauchy(X6) ) ).

% cauchyI
tff(fact_6447_cauchy__def,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
    <=> ! [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
         => ? [K3: nat] :
            ! [M6: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),M6))
             => ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
                 => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X6,M6)),aa(nat,rat,X6,N5)))),R5)) ) ) ) ) ).

% cauchy_def
tff(fact_6448_positive__Real,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( pp(aa(real,bool,positive2,real2(X6)))
      <=> ? [R5: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
            & ? [K3: nat] :
              ! [N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,X6,N5))) ) ) ) ) ).

% positive_Real
tff(fact_6449_cauchy__not__vanishes__cases,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B4: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B4))
            & ? [K: nat] :
                ( ! [N4: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N4))
                   => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),B4),aa(rat,rat,uminus_uminus(rat),aa(nat,rat,X6,N4)))) )
                | ! [N4: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N4))
                   => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),B4),aa(nat,rat,X6,N4))) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_6450_cauchy__not__vanishes,axiom,
    ! [X6: fun(nat,rat)] :
      ( cauchy(X6)
     => ( ~ vanishes(X6)
       => ? [B4: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B4))
            & ? [K: nat] :
              ! [N4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N4))
               => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),B4),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N4)))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_6451_vanishesD,axiom,
    ! [X6: fun(nat,rat),R: rat] :
      ( vanishes(X6)
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R))
       => ? [K: nat] :
          ! [N4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N4))
           => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N4))),R)) ) ) ) ).

% vanishesD
tff(fact_6452_vanishesI,axiom,
    ! [X6: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R3))
         => ? [K4: nat] :
            ! [N3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K4),N3))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N3))),R3)) ) )
     => vanishes(X6) ) ).

% vanishesI
tff(fact_6453_vanishes__def,axiom,
    ! [X6: fun(nat,rat)] :
      ( vanishes(X6)
    <=> ! [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
         => ? [K3: nat] :
            ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N5))),R5)) ) ) ) ).

% vanishes_def
tff(fact_6454_vanishes__mult__bounded,axiom,
    ! [X6: fun(nat,rat),Y6: fun(nat,rat)] :
      ( ? [A8: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A8))
          & ! [N3: nat] : pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X6,N3))),A8)) )
     => ( vanishes(Y6)
       => vanishes(aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_afe(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),X6),Y6)) ) ) ).

% vanishes_mult_bounded
tff(fact_6455_surj__int__decode,axiom,
    image2(nat,int,nat_int_decode,top_top(set(nat))) = top_top(set(int)) ).

% surj_int_decode
tff(fact_6456_map__of__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),Xs: list(product_prod(A,C))] : map_of(A,B,map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_lz(fun(C,B),fun(A,fun(C,product_prod(A,B))),F3)),Xs)) = aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F3)),map_of(A,C,Xs)) ).

% map_of_map
tff(fact_6457_map__option__eq__Some,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Xo: option(B),Y: A] :
      ( ( aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F3),Xo) = aa(A,option(A),some(A),Y) )
    <=> ? [Z3: B] :
          ( ( Xo = aa(B,option(B),some(B),Z3) )
          & ( aa(B,A,F3,Z3) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_6458_None__eq__map__option__iff,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X: option(B)] :
      ( ( none(A) = aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F3),X) )
    <=> ( X = none(B) ) ) ).

% None_eq_map_option_iff
tff(fact_6459_map__option__is__None,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Opt: option(B)] :
      ( ( aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F3),Opt) = none(A) )
    <=> ( Opt = none(B) ) ) ).

% map_option_is_None
tff(fact_6460_option_Omap__disc__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: option(A)] :
      ( ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),A3) = none(B) )
    <=> ( A3 = none(A) ) ) ).

% option.map_disc_iff
tff(fact_6461_case__map__option,axiom,
    ! [B: $tType,A: $tType,C: $tType,G3: A,H: fun(B,A),F3: fun(C,B),X: option(C)] : aa(option(B),A,aa(fun(B,A),fun(option(B),A),aa(A,fun(fun(B,A),fun(option(B),A)),case_option(A,B),G3),H),aa(option(C),option(B),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F3),X)) = aa(option(C),A,aa(fun(C,A),fun(option(C),A),aa(A,fun(fun(C,A),fun(option(C),A)),case_option(A,C),G3),aa(fun(C,B),fun(C,A),comp(B,A,C,H),F3)),X) ).

% case_map_option
tff(fact_6462_map__option__o__map__upd,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,B),M2: fun(A,option(C)),A3: A,B2: C] : aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F3)),fun_upd(A,option(C),M2,A3,aa(C,option(C),some(C),B2))) = fun_upd(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F3)),M2),A3,aa(B,option(B),some(B),aa(C,B,F3,B2))) ).

% map_option_o_map_upd
tff(fact_6463_bij__int__decode,axiom,
    bij_betw(nat,int,nat_int_decode,top_top(set(nat)),top_top(set(int))) ).

% bij_int_decode
tff(fact_6464_inj__int__decode,axiom,
    ! [A6: set(nat)] : inj_on(nat,int,nat_int_decode,A6) ).

% inj_int_decode
tff(fact_6465_option_Omap__id,axiom,
    ! [A: $tType,T2: option(A)] : aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),id(A)),T2) = T2 ).

% option.map_id
tff(fact_6466_option_Omap__id0,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),id(A)) = id(option(A)) ).

% option.map_id0
tff(fact_6467_map__option_Oidentity,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),aTP_Lamp_abo(A,A)) = id(option(A)) ).

% map_option.identity
tff(fact_6468_map__option_Ocomp,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C),G3: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),option(C)),comp(option(B),option(C),option(A),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),F3)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G3)) = aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,F3),G3)) ).

% map_option.comp
tff(fact_6469_option_Omap__comp,axiom,
    ! [B: $tType,C: $tType,A: $tType,G3: fun(B,C),F3: fun(A,B),V3: option(A)] : aa(option(B),option(C),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),G3),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),V3)) = aa(option(A),option(C),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G3),F3)),V3) ).

% option.map_comp
tff(fact_6470_map__option_Ocompositionality,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(B,C),G3: fun(A,B),Option: option(A)] : aa(option(B),option(C),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),F3),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G3),Option)) = aa(option(A),option(C),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,F3),G3)),Option) ).

% map_option.compositionality
tff(fact_6471_option_Omap__sel,axiom,
    ! [B: $tType,A: $tType,A3: option(A),F3: fun(A,B)] :
      ( ( A3 != none(A) )
     => ( aa(option(B),B,the2(B),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),A3)) = aa(A,B,F3,aa(option(A),A,the2(A),A3)) ) ) ).

% option.map_sel
tff(fact_6472_option_Omap__ident,axiom,
    ! [A: $tType,T2: option(A)] : aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),aTP_Lamp_abo(A,A)),T2) = T2 ).

% option.map_ident
tff(fact_6473_int__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,int,nat_int_decode,X) = aa(nat,int,nat_int_decode,Y) )
    <=> ( X = Y ) ) ).

% int_decode_eq
tff(fact_6474_map__option__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y: option(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( X = Y )
     => ( ! [A5: A] :
            ( ( Y = aa(A,option(A),some(A),A5) )
           => ( aa(A,B,F3,A5) = aa(A,B,G3,A5) ) )
       => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G3),Y) ) ) ) ).

% map_option_cong
tff(fact_6475_option_Osimps_I9_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X2: A] : aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),aa(A,option(A),some(A),X2)) = aa(B,option(B),some(B),aa(A,B,F3,X2)) ).

% option.simps(9)
tff(fact_6476_option_Osimps_I8_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B)] : aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),none(A)) = none(B) ).

% option.simps(8)
tff(fact_6477_option_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,nat),G3: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),nat),comp(option(B),nat,option(A),size_option(B,F3)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G3)) = size_option(A,aa(fun(A,B),fun(A,nat),comp(B,nat,A,F3),G3)) ).

% option.size_gen_o_map
tff(fact_6478_option_Oinj__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => inj_on(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),top_top(set(option(A)))) ) ).

% option.inj_map
tff(fact_6479_map__option__case,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Y: option(B)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F3),Y) = aa(option(B),option(A),aa(fun(B,option(A)),fun(option(B),option(A)),aa(option(A),fun(fun(B,option(A)),fun(option(B),option(A))),case_option(option(A),B),none(A)),aTP_Lamp_afg(fun(B,A),fun(B,option(A)),F3)),Y) ).

% map_option_case
tff(fact_6480_nat__to__rat__surj__def,axiom,
    ! [N: nat] : nat_to_rat_surj(N) = aa(product_prod(nat,nat),rat,aa(fun(nat,fun(nat,rat)),fun(product_prod(nat,nat),rat),product_case_prod(nat,nat,rat),aTP_Lamp_afh(nat,fun(nat,rat))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ).

% nat_to_rat_surj_def
tff(fact_6481_aboveS__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_aboveS(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_afi(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).

% aboveS_def
tff(fact_6482_option_Orec__o__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G3: C,Ga: fun(B,C),F3: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),C),comp(option(B),C,option(A),aa(fun(B,C),fun(option(B),C),aa(C,fun(fun(B,C),fun(option(B),C)),rec_option(C,B),G3),Ga)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3)) = aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),rec_option(C,A),G3),aa(fun(A,B),fun(A,C),aTP_Lamp_yd(fun(B,C),fun(fun(A,B),fun(A,C)),Ga),F3)) ).

% option.rec_o_map
tff(fact_6483_bij__int__encode,axiom,
    bij_betw(int,nat,nat_int_encode,top_top(set(int)),top_top(set(nat))) ).

% bij_int_encode
tff(fact_6484_int__encode__inverse,axiom,
    ! [X: int] : aa(nat,int,nat_int_decode,aa(int,nat,nat_int_encode,X)) = X ).

% int_encode_inverse
tff(fact_6485_int__decode__inverse,axiom,
    ! [N: nat] : aa(int,nat,nat_int_encode,aa(nat,int,nat_int_decode,N)) = N ).

% int_decode_inverse
tff(fact_6486_inj__int__encode,axiom,
    ! [A6: set(int)] : inj_on(int,nat,nat_int_encode,A6) ).

% inj_int_encode
tff(fact_6487_int__encode__eq,axiom,
    ! [X: int,Y: int] :
      ( ( aa(int,nat,nat_int_encode,X) = aa(int,nat,nat_int_encode,Y) )
    <=> ( X = Y ) ) ).

% int_encode_eq
tff(fact_6488_option_Osimps_I7_J,axiom,
    ! [C: $tType,A: $tType,F1: C,F22: fun(A,C),X2: A] : aa(option(A),C,aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),rec_option(C,A),F1),F22),aa(A,option(A),some(A),X2)) = aa(A,C,F22,X2) ).

% option.simps(7)
tff(fact_6489_option_Osimps_I6_J,axiom,
    ! [A: $tType,C: $tType,F1: C,F22: fun(A,C)] : aa(option(A),C,aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),rec_option(C,A),F1),F22),none(A)) = F1 ).

% option.simps(6)
tff(fact_6490_surj__int__encode,axiom,
    image2(int,nat,nat_int_encode,top_top(set(int))) = top_top(set(nat)) ).

% surj_int_encode
tff(fact_6491_relInvImage__def,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R2: set(product_prod(B,B)),F3: fun(A,B)] : bNF_Gr7122648621184425601vImage(A,B,A6,R2,F3) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),bool),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool)),aTP_Lamp_afj(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),A6),R2),F3)) ).

% relInvImage_def
tff(fact_6492_scomp__unfold,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,X5: fun(A,product_prod(B,C)),Xa: fun(B,fun(C,D)),Xb3: A] : aa(A,D,product_scomp(A,B,C,D,X5,Xa),Xb3) = aa(C,D,aa(B,fun(C,D),Xa,aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),X5,Xb3))),aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),X5,Xb3))) ).

% scomp_unfold
tff(fact_6493_scomp__apply,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: fun(B,product_prod(C,D)),G3: fun(C,fun(D,A)),X: B] : aa(B,A,product_scomp(B,C,D,A,F3,G3),X) = aa(product_prod(C,D),A,aa(fun(C,fun(D,A)),fun(product_prod(C,D),A),product_case_prod(C,D,A),G3),aa(B,product_prod(C,D),F3,X)) ).

% scomp_apply
tff(fact_6494_scomp__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: fun(A,product_prod(B,C))] : product_scomp(A,B,C,product_prod(B,C),X,product_Pair(B,C)) = X ).

% scomp_Pair
tff(fact_6495_Pair__scomp,axiom,
    ! [A: $tType,B: $tType,C: $tType,X: C,F3: fun(C,fun(A,B))] : product_scomp(A,C,A,B,aa(C,fun(A,product_prod(C,A)),product_Pair(C,A),X),F3) = aa(C,fun(A,B),F3,X) ).

% Pair_scomp
tff(fact_6496_scomp__scomp,axiom,
    ! [A: $tType,E: $tType,F: $tType,B: $tType,D: $tType,C: $tType,F3: fun(A,product_prod(E,F)),G3: fun(E,fun(F,product_prod(C,D))),H: fun(C,fun(D,B))] : product_scomp(A,C,D,B,product_scomp(A,E,F,product_prod(C,D),F3,G3),H) = product_scomp(A,E,F,B,F3,aa(fun(C,fun(D,B)),fun(E,fun(F,B)),aTP_Lamp_afk(fun(E,fun(F,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F,B))),G3),H)) ).

% scomp_scomp
tff(fact_6497_scomp__def,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,F3: fun(A,product_prod(B,C)),G3: fun(B,fun(C,D)),X5: A] : aa(A,D,product_scomp(A,B,C,D,F3,G3),X5) = aa(product_prod(B,C),D,aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),G3),aa(A,product_prod(B,C),F3,X5)) ).

% scomp_def
tff(fact_6498_prod__list__def,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( groups5270119922927024881d_list(A) = groups_monoid_F(A,times_times(A),one_one(A)) ) ) ).

% prod_list_def
tff(fact_6499_Zfun__imp__Zfun,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,C),K5: real] :
          ( zfun(A,B,F3,F4)
         => ( eventually(A,aa(real,fun(A,bool),aa(fun(A,C),fun(real,fun(A,bool)),aTP_Lamp_vx(fun(A,B),fun(fun(A,C),fun(real,fun(A,bool))),F3),G3),K5),F4)
           => zfun(A,C,G3,F4) ) ) ) ).

% Zfun_imp_Zfun
tff(fact_6500_Zfun__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F4: filter(A)] : zfun(A,B,aTP_Lamp_afl(A,B),F4) ) ).

% Zfun_zero
tff(fact_6501_Zfun__mult,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(D,A),F4: filter(D),G3: fun(D,A)] :
          ( zfun(D,A,F3,F4)
         => ( zfun(D,A,G3,F4)
           => zfun(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_sa(fun(D,A),fun(fun(D,A),fun(D,A)),F3),G3),F4) ) ) ) ).

% Zfun_mult
tff(fact_6502_Zfun__mult__left,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(D,A),F4: filter(D),A3: A] :
          ( zfun(D,A,F3,F4)
         => zfun(D,A,aa(A,fun(D,A),aTP_Lamp_rz(fun(D,A),fun(A,fun(D,A)),F3),A3),F4) ) ) ).

% Zfun_mult_left
tff(fact_6503_Zfun__mult__right,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(D,A),F4: filter(D),A3: A] :
          ( zfun(D,A,F3,F4)
         => zfun(D,A,aa(A,fun(D,A),aTP_Lamp_ry(fun(D,A),fun(A,fun(D,A)),F3),A3),F4) ) ) ).

% Zfun_mult_right
tff(fact_6504_sum__list__def,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( groups8242544230860333062m_list(A) = groups_monoid_F(A,plus_plus(A),zero_zero(A)) ) ) ).

% sum_list_def
tff(fact_6505_decseq__ge,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),L5: A,N: nat] :
          ( order_antimono(nat,A,X6)
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L5),aa(nat,A,X6,N))) ) ) ) ).

% decseq_ge
tff(fact_6506_not__in__connected__cases,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A),X: A] :
          ( topolo1966860045006549960nected(A,S3)
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S3))
           => ( ( S3 != bot_bot(set(A)) )
             => ( ( pp(aa(set(A),bool,condit941137186595557371_above(A),S3))
                 => ~ ! [Y4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),S3))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) ) )
               => ~ ( pp(aa(set(A),bool,condit1013018076250108175_below(A),S3))
                   => ~ ! [Y4: A] :
                          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),S3))
                         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) ) ) ) ) ) ) ) ).

% not_in_connected_cases
tff(fact_6507_decseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),I2: nat,J2: nat] :
          ( order_antimono(nat,A,F3)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,J2)),aa(nat,A,F3,I2))) ) ) ) ).

% decseqD
tff(fact_6508_decseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( order_antimono(nat,A,X6)
        <=> ! [M6: nat,N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M6),N5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N5)),aa(nat,A,X6,M6))) ) ) ) ).

% decseq_def
tff(fact_6509_connected__iff__interval,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [U3: set(A)] :
          ( topolo1966860045006549960nected(A,U3)
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),U3))
             => ! [Xa3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),U3))
                 => ! [Z3: A] :
                      ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Z3))
                     => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z3),Xa3))
                       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),U3)) ) ) ) ) ) ) ).

% connected_iff_interval
tff(fact_6510_connectedI__interval,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [U3: set(A)] :
          ( ! [X3: A,Y3: A,Z: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),U3))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),U3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),Y3))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),U3)) ) ) ) )
         => topolo1966860045006549960nected(A,U3) ) ) ).

% connectedI_interval
tff(fact_6511_connectedD__interval,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [U3: set(A),X: A,Y: A,Z2: A] :
          ( topolo1966860045006549960nected(A,U3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),U3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),U3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),U3)) ) ) ) ) ) ) ).

% connectedD_interval
tff(fact_6512_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_antimono(A,B,F3)
        <=> ! [X4: A,Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y5)),aa(A,B,F3,X4))) ) ) ) ).

% antimono_def
tff(fact_6513_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y3)),aa(A,B,F3,X3))) )
         => order_antimono(A,B,F3) ) ) ).

% antimonoI
tff(fact_6514_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y)),aa(A,B,F3,X))) ) ) ) ).

% antimonoE
tff(fact_6515_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y)),aa(A,B,F3,X))) ) ) ) ).

% antimonoD
tff(fact_6516_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A6: fun(nat,A),I2: nat] :
          ( order_antimono(nat,A,A6)
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A6,aa(nat,nat,suc,I2))),aa(nat,A,A6,I2))) ) ) ).

% decseq_SucD
tff(fact_6517_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N3))),aa(nat,A,X6,N3)))
         => order_antimono(nat,A,X6) ) ) ).

% decseq_SucI
tff(fact_6518_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( order_antimono(nat,A,F3)
        <=> ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N5))),aa(nat,A,F3,N5))) ) ) ).

% decseq_Suc_iff
tff(fact_6519_antimono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Q: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),Q))
         => order_antimono(nat,A,aTP_Lamp_afm(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_6520_independent__span__bound,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [T5: set(A),S3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),T5))
         => ( ~ real_V358717886546972837endent(A,S3)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S3),real_Vector_span(A,T5)))
             => ( pp(aa(set(A),bool,finite_finite2(A),S3))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),S3)),aa(set(A),nat,finite_card(A),T5))) ) ) ) ) ) ).

% independent_span_bound
tff(fact_6521_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),K2: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K2)),F3),bot_bot(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),bot_bot(A)) )
           => ( complete_lattice_lfp(A,F3) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),bot_bot(A)) ) ) ) ) ).

% lfp_Kleene_iter
tff(fact_6522_span__insert__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] : real_Vector_span(A,aa(set(A),set(A),insert(A,zero_zero(A)),S3)) = real_Vector_span(A,S3) ) ).

% span_insert_0
tff(fact_6523_span__empty,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ( real_Vector_span(A,bot_bot(set(A))) = aa(set(A),set(A),insert(A,zero_zero(A)),bot_bot(set(A))) ) ) ).

% span_empty
tff(fact_6524_span__delete__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] : real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,zero_zero(A)),bot_bot(set(A))))) = real_Vector_span(A,S3) ) ).

% span_delete_0
tff(fact_6525_span__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),real_Vector_span(A,S3))) ) ).

% span_0
tff(fact_6526_span__induct__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,S3: set(A),H: fun(A,bool)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,S3)))
         => ( pp(aa(A,bool,H,zero_zero(A)))
           => ( ! [C2: real,X3: A,Y3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
                 => ( pp(aa(A,bool,H,Y3))
                   => pp(aa(A,bool,H,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C2),X3)),Y3))) ) )
             => pp(aa(A,bool,H,X)) ) ) ) ) ).

% span_induct_alt
tff(fact_6527_lfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( complete_lattice_lfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F3)) = complete_lattice_lfp(A,F3) ) ) ) ).

% lfp_funpow
tff(fact_6528_lfp__induct2,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,F3: fun(set(product_prod(A,B)),set(product_prod(A,B))),P2: fun(A,fun(B,bool))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),complete_lattice_lfp(set(product_prod(A,B)),F3)))
     => ( pp(aa(fun(set(product_prod(A,B)),set(product_prod(A,B))),bool,order_mono(set(product_prod(A,B)),set(product_prod(A,B))),F3))
       => ( ! [A5: A,B4: B] :
              ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(set(product_prod(A,B)),set(product_prod(A,B)),F3,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),complete_lattice_lfp(set(product_prod(A,B)),F3)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2))))))
             => pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P2,A3),B2)) ) ) ) ).

% lfp_induct2
tff(fact_6529_lfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),P2: fun(A,bool)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( ! [S4: A] :
                ( pp(aa(A,bool,P2,S4))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),S4),complete_lattice_lfp(A,F3)))
                 => pp(aa(A,bool,P2,aa(A,A,F3,S4))) ) )
           => ( ! [M8: set(A)] :
                  ( ! [X5: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),M8))
                     => pp(aa(A,bool,P2,X5)) )
                 => pp(aa(A,bool,P2,aa(set(A),A,complete_Sup_Sup(A),M8))) )
             => pp(aa(A,bool,P2,complete_lattice_lfp(A,F3))) ) ) ) ) ).

% lfp_ordinal_induct
tff(fact_6530_def__lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: A,F3: fun(A,A),P2: A] :
          ( ( A6 = complete_lattice_lfp(A,F3) )
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),A6),P2))),P2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),P2)) ) ) ) ) ).

% def_lfp_induct
tff(fact_6531_le__rel__bool__arg__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X6: fun(bool,A),Y6: fun(bool,A)] :
          ( pp(aa(fun(bool,A),bool,aa(fun(bool,A),fun(fun(bool,A),bool),ord_less_eq(fun(bool,A)),X6),Y6))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X6,fFalse)),aa(bool,A,Y6,fFalse)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X6,fTrue)),aa(bool,A,Y6,fTrue))) ) ) ) ).

% le_rel_bool_arg_iff
tff(fact_6532_lfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),G3: fun(A,A)] :
          ( ! [Z7: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,Z7)),aa(A,A,G3,Z7)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F3)),complete_lattice_lfp(A,G3))) ) ) ).

% lfp_mono
tff(fact_6533_lfp__lowerbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),A6: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,A6)),A6))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F3)),A6)) ) ) ).

% lfp_lowerbound
tff(fact_6534_lfp__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),A6: A] :
          ( ! [U4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,U4)),U4))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),U4)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),complete_lattice_lfp(A,F3))) ) ) ).

% lfp_greatest
tff(fact_6535_lfp__lfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,fun(A,A))] :
          ( ! [X3: A,Y3: A,W: A,Z: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),W)),aa(A,A,aa(A,fun(A,A),F3,Y3),Z))) ) )
         => ( complete_lattice_lfp(A,aTP_Lamp_afn(fun(A,fun(A,A)),fun(A,A),F3)) = complete_lattice_lfp(A,aTP_Lamp_afo(fun(A,fun(A,A)),fun(A,A),F3)) ) ) ) ).

% lfp_lfp
tff(fact_6536_lfp__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F4: fun(A,A),X: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F4))
         => ( ( aa(A,A,F4,X) = X )
           => ( ! [Z: A] :
                  ( ( aa(A,A,F4,Z) = Z )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z)) )
             => ( complete_lattice_lfp(A,F4) = X ) ) ) ) ) ).

% lfp_eqI
tff(fact_6537_lfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A)] : complete_lattice_lfp(A,F3) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_afp(fun(A,A),fun(A,bool),F3))) ) ).

% lfp_def
tff(fact_6538_lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),P2: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_lattice_lfp(A,F3)),P2))),P2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F3)),P2)) ) ) ) ).

% lfp_induct
tff(fact_6539_lfp__transfer__bounded,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [P2: fun(A,bool),F3: fun(A,A),Alpha: fun(A,B),G3: fun(B,B)] :
          ( pp(aa(A,bool,P2,bot_bot(A)))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,P2,X3))
               => pp(aa(A,bool,P2,aa(A,A,F3,X3))) )
           => ( ! [M8: fun(nat,A)] :
                  ( ! [I4: nat] : pp(aa(A,bool,P2,aa(nat,A,M8,I4)))
                 => pp(aa(A,bool,P2,aa(set(A),A,complete_Sup_Sup(A),image2(nat,A,M8,top_top(set(nat)))))) )
             => ( ! [M8: fun(nat,A)] :
                    ( pp(aa(fun(nat,A),bool,order_mono(nat,A),M8))
                   => ( ! [I4: nat] : pp(aa(A,bool,P2,aa(nat,A,M8,I4)))
                     => ( aa(A,B,Alpha,aa(set(A),A,complete_Sup_Sup(A),image2(nat,A,M8,top_top(set(nat))))) = aa(set(B),B,complete_Sup_Sup(B),image2(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_afq(fun(A,B),fun(fun(nat,A),fun(nat,B)),Alpha),M8),top_top(set(nat)))) ) ) )
               => ( order_sup_continuous(A,A,F3)
                 => ( order_sup_continuous(B,B,G3)
                   => ( ! [X3: A] :
                          ( pp(aa(A,bool,P2,X3))
                         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),complete_lattice_lfp(A,F3)))
                           => ( aa(A,B,Alpha,aa(A,A,F3,X3)) = aa(B,B,G3,aa(A,B,Alpha,X3)) ) ) )
                     => ( ! [X3: B] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Alpha,bot_bot(A))),aa(B,B,G3,X3)))
                       => ( aa(A,B,Alpha,complete_lattice_lfp(A,F3)) = complete_lattice_lfp(B,G3) ) ) ) ) ) ) ) ) ) ) ).

% lfp_transfer_bounded
tff(fact_6540_dim__le__card,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V: set(A),W3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V),real_Vector_span(A,W3)))
         => ( pp(aa(set(A),bool,finite_finite2(A),W3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),real_Vector_dim(A,V)),aa(set(A),nat,finite_card(A),W3))) ) ) ) ).

% dim_le_card
tff(fact_6541_lfp__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [Alpha: fun(A,B),F3: fun(A,A),G3: fun(B,B)] :
          ( order_sup_continuous(A,B,Alpha)
         => ( order_sup_continuous(A,A,F3)
           => ( order_sup_continuous(B,B,G3)
             => ( ! [X3: B] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Alpha,bot_bot(A))),aa(B,B,G3,X3)))
               => ( ! [X3: A] :
                      ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),complete_lattice_lfp(A,F3)))
                     => ( aa(A,B,Alpha,aa(A,A,F3,X3)) = aa(B,B,G3,aa(A,B,Alpha,X3)) ) )
                 => ( aa(A,B,Alpha,complete_lattice_lfp(A,F3)) = complete_lattice_lfp(B,G3) ) ) ) ) ) ) ) ).

% lfp_transfer
tff(fact_6542_dim__le__card_H,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),real_Vector_dim(A,S2)),aa(set(A),nat,finite_card(A),S2))) ) ) ).

% dim_le_card'
tff(fact_6543_span__card__ge__dim,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B6: set(A),V: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),V))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),V),real_Vector_span(A,B6)))
           => ( pp(aa(set(A),bool,finite_finite2(A),B6))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),real_Vector_dim(A,V)),aa(set(A),nat,finite_card(A),B6))) ) ) ) ) ).

% span_card_ge_dim
tff(fact_6544_dim__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V: set(A)] :
          ( ( ? [B7: set(A)] :
                ( ~ real_V358717886546972837endent(A,B7)
                & ( real_Vector_span(A,B7) = real_Vector_span(A,V) ) )
           => ( real_Vector_dim(A,V) = aa(set(A),nat,finite_card(A),fChoice(set(A),aTP_Lamp_afr(set(A),fun(set(A),bool),V))) ) )
          & ( ~ ? [B4: set(A)] :
                  ( ~ real_V358717886546972837endent(A,B4)
                  & ( real_Vector_span(A,B4) = real_Vector_span(A,V) ) )
           => ( real_Vector_dim(A,V) = zero_zero(nat) ) ) ) ) ).

% dim_def
tff(fact_6545_linear__indep__image__lemma,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F3: fun(A,B),B6: set(A),X: A] :
          ( real_Vector_linear(A,B,F3)
         => ( pp(aa(set(A),bool,finite_finite2(A),B6))
           => ( ~ real_V358717886546972837endent(B,image2(A,B,F3,B6))
             => ( inj_on(A,B,F3,B6)
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,B6)))
                 => ( ( aa(A,B,F3,X) = zero_zero(B) )
                   => ( X = zero_zero(A) ) ) ) ) ) ) ) ) ).

% linear_indep_image_lemma
tff(fact_6546_some__equality,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A] :
      ( pp(aa(A,bool,P2,A3))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
           => ( X3 = A3 ) )
       => ( fChoice(A,P2) = A3 ) ) ) ).

% some_equality
tff(fact_6547_some__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aTP_Lamp_afs(A,fun(A,bool),X)) = X ).

% some_eq_trivial
tff(fact_6548_some__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aa(A,fun(A,bool),fequal(A),X)) = X ).

% some_sym_eq_trivial
tff(fact_6549_linear__eq__0__on__span,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [F3: fun(A,B),B2: set(A),X: A] :
          ( real_Vector_linear(A,B,F3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B2))
               => ( aa(A,B,F3,X3) = zero_zero(B) ) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),real_Vector_span(A,B2)))
             => ( aa(A,B,F3,X) = zero_zero(B) ) ) ) ) ) ).

% linear_eq_0_on_span
tff(fact_6550_some__in__eq,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),fChoice(A,aTP_Lamp_a(set(A),fun(A,bool),A6))),A6))
    <=> ( A6 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_6551_someI,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A] :
      ( pp(aa(A,bool,P2,X))
     => pp(aa(A,bool,P2,fChoice(A,P2))) ) ).

% someI
tff(fact_6552_Eps__cong,axiom,
    ! [A: $tType,P2: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P2,X3))
        <=> pp(aa(A,bool,Q,X3)) )
     => ( fChoice(A,P2) = fChoice(A,Q) ) ) ).

% Eps_cong
tff(fact_6553_tfl__some,axiom,
    ! [A: $tType,P7: fun(A,bool),X5: A] :
      ( pp(aa(A,bool,P7,X5))
     => pp(aa(A,bool,P7,fChoice(A,P7))) ) ).

% tfl_some
tff(fact_6554_some__eq__imp,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A,B2: A] :
      ( ( fChoice(A,P2) = A3 )
     => ( pp(aa(A,bool,P2,B2))
       => pp(aa(A,bool,P2,A3)) ) ) ).

% some_eq_imp
tff(fact_6555_someI2,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A,Q: fun(A,bool)] :
      ( pp(aa(A,bool,P2,A3))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
           => pp(aa(A,bool,Q,X3)) )
       => pp(aa(A,bool,Q,fChoice(A,P2))) ) ) ).

% someI2
tff(fact_6556_someI__ex,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( ? [X_1: A] : pp(aa(A,bool,P2,X_1))
     => pp(aa(A,bool,P2,fChoice(A,P2))) ) ).

% someI_ex
tff(fact_6557_someI2__ex,axiom,
    ! [A: $tType,P2: fun(A,bool),Q: fun(A,bool)] :
      ( ? [X_1: A] : pp(aa(A,bool,P2,X_1))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P2,X3))
           => pp(aa(A,bool,Q,X3)) )
       => pp(aa(A,bool,Q,fChoice(A,P2))) ) ) ).

% someI2_ex
tff(fact_6558_someI2__bex,axiom,
    ! [A: $tType,A6: set(A),P2: fun(A,bool),Q: fun(A,bool)] :
      ( ? [X5: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A6))
          & pp(aa(A,bool,P2,X5)) )
     => ( ! [X3: A] :
            ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
              & pp(aa(A,bool,P2,X3)) )
           => pp(aa(A,bool,Q,X3)) )
       => pp(aa(A,bool,Q,fChoice(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aft(set(A),fun(fun(A,bool),fun(A,bool)),A6),P2)))) ) ) ).

% someI2_bex
tff(fact_6559_some__eq__ex,axiom,
    ! [A: $tType,P2: fun(A,bool)] :
      ( pp(aa(A,bool,P2,fChoice(A,P2)))
    <=> ? [X_13: A] : pp(aa(A,bool,P2,X_13)) ) ).

% some_eq_ex
tff(fact_6560_some1__equality,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A] :
      ( ? [X5: A] :
          ( pp(aa(A,bool,P2,X5))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P2,Y3))
             => ( Y3 = X5 ) ) )
     => ( pp(aa(A,bool,P2,A3))
       => ( fChoice(A,P2) = A3 ) ) ) ).

% some1_equality
tff(fact_6561_linear__times__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A3: A] : real_Vector_linear(real,A,aTP_Lamp_afu(A,fun(real,A),A3)) ) ).

% linear_times_of_real
tff(fact_6562_linear__scale__real,axiom,
    ! [F3: fun(real,real),R: real,B2: real] :
      ( real_Vector_linear(real,real,F3)
     => ( aa(real,real,F3,aa(real,real,aa(real,fun(real,real),times_times(real),R),B2)) = aa(real,real,aa(real,fun(real,real),times_times(real),R),aa(real,real,F3,B2)) ) ) ).

% linear_scale_real
tff(fact_6563_real__linearD,axiom,
    ! [F3: fun(real,real)] :
      ( real_Vector_linear(real,real,F3)
     => ~ ! [C2: real] : F3 != aa(real,fun(real,real),times_times(real),C2) ) ).

% real_linearD
tff(fact_6564_linear__times,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [C3: A] : real_Vector_linear(A,A,aa(A,fun(A,A),times_times(A),C3)) ) ).

% linear_times
tff(fact_6565_linear__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [F3: fun(A,B)] :
          ( real_Vector_linear(A,B,F3)
         => ( aa(A,B,F3,zero_zero(A)) = zero_zero(B) ) ) ) ).

% linear_0
tff(fact_6566_module__hom__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => real_Vector_linear(A,B,aTP_Lamp_afv(A,B)) ) ).

% module_hom_zero
tff(fact_6567_exE__some,axiom,
    ! [A: $tType,P2: fun(A,bool),C3: A] :
      ( ? [X_1: A] : pp(aa(A,bool,P2,X_1))
     => ( ( C3 = fChoice(A,P2) )
       => pp(aa(A,bool,P2,C3)) ) ) ).

% exE_some
tff(fact_6568_linear__injective__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_Vector_linear(A,B,F3)
         => ( inj_on(A,B,F3,top_top(set(A)))
          <=> ! [X4: A] :
                ( ( aa(A,B,F3,X4) = zero_zero(B) )
               => ( X4 = zero_zero(A) ) ) ) ) ) ).

% linear_injective_0
tff(fact_6569_representation__scale,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V3: A,R: real] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V3),real_Vector_span(A,Basis)))
           => ! [X5: A] : real_V7696804695334737415tation(A,Basis,aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),R),V3),X5) = aa(real,real,aa(real,fun(real,real),times_times(real),R),real_V7696804695334737415tation(A,Basis,V3,X5)) ) ) ) ).

% representation_scale
tff(fact_6570_gfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),K2: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K2)),F3),top_top(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),top_top(A)) )
           => ( complete_lattice_gfp(A,F3) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),top_top(A)) ) ) ) ) ).

% gfp_Kleene_iter
tff(fact_6571_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_ir(A,fun(B,fun(A,fun(B,bool))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% Eps_case_prod_eq
tff(fact_6572_gfp__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F4: fun(A,A),X: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F4))
         => ( ( aa(A,A,F4,X) = X )
           => ( ! [Z: A] :
                  ( ( aa(A,A,F4,Z) = Z )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X)) )
             => ( complete_lattice_gfp(A,F4) = X ) ) ) ) ) ).

% gfp_eqI
tff(fact_6573_gfp__gfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,fun(A,A))] :
          ( ! [X3: A,Y3: A,W: A,Z: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),W)),aa(A,A,aa(A,fun(A,A),F3,Y3),Z))) ) )
         => ( complete_lattice_gfp(A,aTP_Lamp_afw(fun(A,fun(A,A)),fun(A,A),F3)) = complete_lattice_gfp(A,aTP_Lamp_afo(fun(A,fun(A,A)),fun(A,A),F3)) ) ) ) ).

% gfp_gfp
tff(fact_6574_gfp__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),X6: A] :
          ( ! [U4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U4),aa(A,A,F3,U4)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U4),X6)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_gfp(A,F3)),X6)) ) ) ).

% gfp_least
tff(fact_6575_gfp__upperbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X6: A,F3: fun(A,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F3,X6)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),complete_lattice_gfp(A,F3))) ) ) ).

% gfp_upperbound
tff(fact_6576_gfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),G3: fun(A,A)] :
          ( ! [Z7: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,Z7)),aa(A,A,G3,Z7)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_gfp(A,F3)),complete_lattice_gfp(A,G3))) ) ) ).

% gfp_mono
tff(fact_6577_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,P2: fun(product_prod(A,B),bool)] : fChoice(product_prod(A,B),P2) = fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_afx(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),P2))) ).

% split_paired_Eps
tff(fact_6578_representation__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),X5: A] : real_V7696804695334737415tation(A,Basis,zero_zero(A),X5) = zero_zero(real) ) ).

% representation_zero
tff(fact_6579_gfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A)] : complete_lattice_gfp(A,F3) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_afy(fun(A,A),fun(A,bool),F3))) ) ).

% gfp_def
tff(fact_6580_coinduct__lemma,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X6: A,F3: fun(A,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F3)))))
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F3))),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F3))))) ) ) ) ).

% coinduct_lemma
tff(fact_6581_def__coinduct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A6: A,F3: fun(A,A),X6: A] :
          ( ( A6 = complete_lattice_gfp(A,F3) )
         => ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),A6))))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),A6)) ) ) ) ) ).

% def_coinduct
tff(fact_6582_coinduct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),X6: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F3)))))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),complete_lattice_gfp(A,F3))) ) ) ) ).

% coinduct
tff(fact_6583_gfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),P2: fun(A,bool)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( ! [S4: A] :
                ( pp(aa(A,bool,P2,S4))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_gfp(A,F3)),S4))
                 => pp(aa(A,bool,P2,aa(A,A,F3,S4))) ) )
           => ( ! [M8: set(A)] :
                  ( ! [X5: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),M8))
                     => pp(aa(A,bool,P2,X5)) )
                 => pp(aa(A,bool,P2,aa(set(A),A,complete_Inf_Inf(A),M8))) )
             => pp(aa(A,bool,P2,complete_lattice_gfp(A,F3))) ) ) ) ) ).

% gfp_ordinal_induct
tff(fact_6584_gfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),N: nat] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( complete_lattice_gfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F3)) = complete_lattice_gfp(A,F3) ) ) ) ).

% gfp_funpow
tff(fact_6585_Eps__case__prod,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool))] : fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2)) = fChoice(product_prod(A,B),aTP_Lamp_kb(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),P2)) ).

% Eps_case_prod
tff(fact_6586_lfp__le__gfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A)] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F3)),complete_lattice_gfp(A,F3))) ) ) ).

% lfp_le_gfp
tff(fact_6587_gfp__transfer__bounded,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [P2: fun(A,bool),F3: fun(A,A),Alpha: fun(A,B),G3: fun(B,B)] :
          ( pp(aa(A,bool,P2,aa(A,A,F3,top_top(A))))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,P2,X3))
               => pp(aa(A,bool,P2,aa(A,A,F3,X3))) )
           => ( ! [M8: fun(nat,A)] :
                  ( order_antimono(nat,A,M8)
                 => ( ! [I4: nat] : pp(aa(A,bool,P2,aa(nat,A,M8,I4)))
                   => pp(aa(A,bool,P2,aa(set(A),A,complete_Inf_Inf(A),image2(nat,A,M8,top_top(set(nat)))))) ) )
             => ( ! [M8: fun(nat,A)] :
                    ( order_antimono(nat,A,M8)
                   => ( ! [I4: nat] : pp(aa(A,bool,P2,aa(nat,A,M8,I4)))
                     => ( aa(A,B,Alpha,aa(set(A),A,complete_Inf_Inf(A),image2(nat,A,M8,top_top(set(nat))))) = aa(set(B),B,complete_Inf_Inf(B),image2(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_afq(fun(A,B),fun(fun(nat,A),fun(nat,B)),Alpha),M8),top_top(set(nat)))) ) ) )
               => ( order_inf_continuous(A,A,F3)
                 => ( order_inf_continuous(B,B,G3)
                   => ( ! [X3: A] :
                          ( pp(aa(A,bool,P2,X3))
                         => ( aa(A,B,Alpha,aa(A,A,F3,X3)) = aa(B,B,G3,aa(A,B,Alpha,X3)) ) )
                     => ( ! [X3: B] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,G3,X3)),aa(A,B,Alpha,aa(A,A,F3,top_top(A)))))
                       => ( aa(A,B,Alpha,complete_lattice_gfp(A,F3)) = complete_lattice_gfp(B,G3) ) ) ) ) ) ) ) ) ) ) ).

% gfp_transfer_bounded
tff(fact_6588_list__ex__length,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] :
      ( list_ex(A,P2,Xs)
    <=> ? [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),N5))) ) ) ).

% list_ex_length
tff(fact_6589_max__ext_Ocases,axiom,
    ! [A: $tType,A12: set(A),A23: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A12),A23)),max_ext(A,R2)))
     => ~ ( pp(aa(set(A),bool,finite_finite2(A),A12))
         => ( pp(aa(set(A),bool,finite_finite2(A),A23))
           => ( ( A23 != bot_bot(set(A)) )
             => ~ ! [X5: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A12))
                   => ? [Xa4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A23))
                        & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa4)),R2)) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_6590_max__ext_Osimps,axiom,
    ! [A: $tType,A12: set(A),A23: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A12),A23)),max_ext(A,R2)))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),A12))
        & pp(aa(set(A),bool,finite_finite2(A),A23))
        & ( A23 != bot_bot(set(A)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A12))
           => ? [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),A23))
                & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R2)) ) ) ) ) ).

% max_ext.simps
tff(fact_6591_max__ext_Omax__extI,axiom,
    ! [A: $tType,X6: set(A),Y6: set(A),R2: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),Y6))
       => ( ( Y6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => ? [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),Y6))
                    & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa)),R2)) ) )
           => pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X6),Y6)),max_ext(A,R2))) ) ) ) ) ).

% max_ext.max_extI
tff(fact_6592_max__ext__def,axiom,
    ! [A: $tType,X5: set(product_prod(A,A))] : max_ext(A,X5) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),max_extp(A,aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),X5)))) ).

% max_ext_def
tff(fact_6593_max__extp__max__ext__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X5: set(A),Xa: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R2)),X5),Xa))
    <=> pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X5),Xa)),max_ext(A,R2))) ) ).

% max_extp_max_ext_eq
tff(fact_6594_max__ext__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : max_ext(A,R2) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_afz(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R2))) ).

% max_ext_eq
tff(fact_6595_finite__sequence__to__countable__set,axiom,
    ! [A: $tType,X6: set(A)] :
      ( countable_countable(A,X6)
     => ~ ! [F8: fun(nat,set(A))] :
            ( ! [I4: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F8,I4)),X6))
           => ( ! [I4: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F8,I4)),aa(nat,set(A),F8,aa(nat,nat,suc,I4))))
             => ( ! [I4: nat] : pp(aa(set(A),bool,finite_finite2(A),aa(nat,set(A),F8,I4)))
               => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),F8,top_top(set(nat)))) != X6 ) ) ) ) ) ).

% finite_sequence_to_countable_set
tff(fact_6596_ccInf__greatest,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(A),Z2: A] :
          ( countable_countable(A,A6)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(set(A),A,complete_Inf_Inf(A),A6))) ) ) ) ).

% ccInf_greatest
tff(fact_6597_le__ccInf__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(A),B2: A] :
          ( countable_countable(A,A6)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A6)))
          <=> ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X4)) ) ) ) ) ).

% le_ccInf_iff
tff(fact_6598_ccInf__lower2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(A),U: A,V3: A] :
          ( countable_countable(A,A6)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),V3)) ) ) ) ) ).

% ccInf_lower2
tff(fact_6599_ccInf__lower,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(A),X: A] :
          ( countable_countable(A,A6)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),X)) ) ) ) ).

% ccInf_lower
tff(fact_6600_ccInf__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B6: set(A),A6: set(A)] :
          ( countable_countable(A,B6)
         => ( countable_countable(A,A6)
           => ( ! [B4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),B6))
                 => ? [X5: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A6))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),B4)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B6))) ) ) ) ) ).

% ccInf_mono
tff(fact_6601_ccSup__upper2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(A),U: A,V3: A] :
          ( countable_countable(A,A6)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V3),U))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V3),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ) ) ).

% ccSup_upper2
tff(fact_6602_ccSup__le__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(A),B2: A] :
          ( countable_countable(A,A6)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),B2))
          <=> ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2)) ) ) ) ) ).

% ccSup_le_iff
tff(fact_6603_ccSup__upper,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(A),X: A] :
          ( countable_countable(A,A6)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ) ).

% ccSup_upper
tff(fact_6604_ccSup__least,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(A),Z2: A] :
          ( countable_countable(A,A6)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),Z2)) ) ) ) ).

% ccSup_least
tff(fact_6605_ccSup__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B6: set(A),A6: set(A)] :
          ( countable_countable(A,B6)
         => ( countable_countable(A,A6)
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A6))
                 => ? [X5: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),B6))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X5)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B6))) ) ) ) ) ).

% ccSup_mono
tff(fact_6606_ccSup__subset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B6: set(A),A6: set(A)] :
          ( countable_countable(A,B6)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B6))) ) ) ) ).

% ccSup_subset_mono
tff(fact_6607_ccInf__superset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(A),B6: set(A)] :
          ( countable_countable(A,A6)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B6))) ) ) ) ).

% ccInf_superset_mono
tff(fact_6608_ccSUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(B),I2: B,U: A,F3: fun(B,A)] :
          ( countable_countable(B,A6)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,I2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6)))) ) ) ) ) ).

% ccSUP_upper2
tff(fact_6609_ccSUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(B),F3: fun(B,A),U: A] :
          ( countable_countable(B,A6)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),U))
          <=> ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X4)),U)) ) ) ) ) ).

% ccSUP_le_iff
tff(fact_6610_ccSUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(B),I2: B,F3: fun(B,A)] :
          ( countable_countable(B,A6)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6)))) ) ) ) ).

% ccSUP_upper
tff(fact_6611_ccSUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(B),F3: fun(B,A),U: A] :
          ( countable_countable(B,A6)
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I3)),U)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),U)) ) ) ) ).

% ccSUP_least
tff(fact_6612_ccSUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(B),B6: set(C),F3: fun(B,A),G3: fun(C,A)] :
          ( countable_countable(B,A6)
         => ( countable_countable(C,B6)
           => ( ! [N3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),N3),A6))
                 => ? [X5: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X5),B6))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,N3)),aa(C,A,G3,X5))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Sup_Sup(A),image2(C,A,G3,B6)))) ) ) ) ) ).

% ccSUP_mono
tff(fact_6613_ccINF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(B),U: A,F3: fun(B,A)] :
          ( countable_countable(B,A6)
         => ( ! [I3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,I3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6)))) ) ) ) ).

% ccINF_greatest
tff(fact_6614_le__ccINF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(B),U: A,F3: fun(B,A)] :
          ( countable_countable(B,A6)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))))
          <=> ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,X4))) ) ) ) ) ).

% le_ccINF_iff
tff(fact_6615_ccINF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(B),I2: B,F3: fun(B,A),U: A] :
          ( countable_countable(B,A6)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A6))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),U))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),U)) ) ) ) ) ).

% ccINF_lower2
tff(fact_6616_ccINF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(B),I2: B,F3: fun(B,A)] :
          ( countable_countable(B,A6)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),aa(B,A,F3,I2))) ) ) ) ).

% ccINF_lower
tff(fact_6617_ccINF__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(B),B6: set(C),F3: fun(B,A),G3: fun(C,A)] :
          ( countable_countable(B,A6)
         => ( countable_countable(C,B6)
           => ( ! [M: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),M),B6))
                 => ? [X5: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),A6))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X5)),aa(C,A,G3,M))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Inf_Inf(A),image2(C,A,G3,B6)))) ) ) ) ) ).

% ccINF_mono
tff(fact_6618_ccSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(A),B6: set(A)] :
          ( countable_countable(A,A6)
         => ( countable_countable(A,B6)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A6)),aa(set(A),A,complete_Sup_Sup(A),B6)))) ) ) ) ).

% ccSup_inter_less_eq
tff(fact_6619_less__eq__ccInf__inter,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(A),B6: set(A)] :
          ( countable_countable(A,A6)
         => ( countable_countable(A,B6)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A6)),aa(set(A),A,complete_Inf_Inf(A),B6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6)))) ) ) ) ).

% less_eq_ccInf_inter
tff(fact_6620_ccSUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B6: set(B),A6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( countable_countable(B,B6)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Sup_Sup(A),image2(B,A,G3,B6)))) ) ) ) ) ).

% ccSUP_subset_mono
tff(fact_6621_ccINF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A6: set(B),B6: set(B),F3: fun(B,A),G3: fun(B,A)] :
          ( countable_countable(B,A6)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),A6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),B6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G3,X3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,F3,A6))),aa(set(A),A,complete_Inf_Inf(A),image2(B,A,G3,B6)))) ) ) ) ) ).

% ccINF_superset_mono
tff(fact_6622_mono__ccSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( countable_countable(A,A6)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image2(A,B,F3,A6))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),A6)))) ) ) ) ).

% mono_ccSup
tff(fact_6623_mono__ccSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F3: fun(A,B),I5: set(C),A6: fun(C,A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( countable_countable(C,I5)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aga(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A6),I5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),image2(C,A,A6,I5))))) ) ) ) ).

% mono_ccSUP
tff(fact_6624_mono__ccInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F3: fun(A,B),A6: set(A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( countable_countable(A,A6)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A6))),aa(set(B),B,complete_Inf_Inf(B),image2(A,B,F3,A6)))) ) ) ) ).

% mono_ccInf
tff(fact_6625_mono__ccINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( counta3822494911875563373attice(B)
        & counta4013691401010221786attice(A) )
     => ! [F3: fun(A,B),I5: set(C),A6: fun(C,A)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( countable_countable(C,I5)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),image2(C,A,A6,I5)))),aa(set(B),B,complete_Inf_Inf(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aga(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A6),I5)))) ) ) ) ).

% mono_ccINF
tff(fact_6626_map__project__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),A6: set(A)] : map_project(A,B,F3,A6) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_agb(fun(A,option(B)),fun(set(A),fun(B,bool)),F3),A6)) ).

% map_project_def
tff(fact_6627_min__ext__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : min_ext(A,R) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_agc(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),R)) ).

% min_ext_def
tff(fact_6628_Chains__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : chains(A,R) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_agd(set(product_prod(A,A)),fun(set(A),bool),R)) ).

% Chains_def
tff(fact_6629_cclfp__lowerbound,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F3: fun(A,A),A6: A] :
          ( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,A6)),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),order_532582986084564980_cclfp(A,F3)),A6)) ) ) ) ).

% cclfp_lowerbound
tff(fact_6630_Chains__subset_H,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R)
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R)))),chains(A,R))) ) ).

% Chains_subset'
tff(fact_6631_Chains__alt__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R)
     => ( chains(A,R) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R))) ) ) ).

% Chains_alt_def
tff(fact_6632_Chains__subset,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),chains(A,R)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R))))) ).

% Chains_subset
tff(fact_6633_Zorns__po__lemma,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_7125193373082350890der_on(A,field2(A,R),R)
     => ( ! [C7: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),C7),chains(A,R)))
           => ? [X5: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),field2(A,R)))
                & ! [Xa4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),C7))
                   => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa4),X5)),R)) ) ) )
       => ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),field2(A,R)))
            & ! [Xa: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),field2(A,R)))
               => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa)),R))
                 => ( Xa = X3 ) ) ) ) ) ) ).

% Zorns_po_lemma
tff(fact_6634_construct__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [B6: set(A),G3: fun(A,B),V3: A] : real_V4425403222259421789struct(A,B,B6,G3,V3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_age(set(A),fun(fun(A,B),fun(A,fun(A,B))),B6),G3),V3)),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_agf(set(A),fun(A,fun(A,bool)),B6),V3))) ) ).

% construct_def
tff(fact_6635_construct__outside,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [B6: set(A),V3: A,F3: fun(A,B)] :
          ( ~ real_V358717886546972837endent(A,B6)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V3),real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),real_V4986007116245087402_basis(A,B6)),B6))))
           => ( real_V4425403222259421789struct(A,B,B6,F3,V3) = zero_zero(B) ) ) ) ) ).

% construct_outside
tff(fact_6636_natLeq__Partial__order,axiom,
    order_7125193373082350890der_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Partial_order
tff(fact_6637_natLeq__Preorder,axiom,
    order_preorder_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq),bNF_Ca8665028551170535155natLeq) ).

% natLeq_Preorder
tff(fact_6638_Range__insert,axiom,
    ! [A: $tType,B: $tType,A3: B,B2: A,R: set(product_prod(B,A))] : range2(B,A,aa(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),A3),B2)),R)) = aa(set(A),set(A),insert(A,B2),range2(B,A,R)) ).

% Range_insert
tff(fact_6639_Range_Ocases,axiom,
    ! [B: $tType,A: $tType,A3: B,R: set(product_prod(A,B))] :
      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),range2(A,B,R)))
     => ~ ! [A5: A] : ~ pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),A3)),R)) ) ).

% Range.cases
tff(fact_6640_Range_Osimps,axiom,
    ! [B: $tType,A: $tType,A3: B,R: set(product_prod(A,B))] :
      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A3),range2(A,B,R)))
    <=> ? [A7: A,B5: B] :
          ( ( A3 = B5 )
          & pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5)),R)) ) ) ).

% Range.simps
tff(fact_6641_Range_Ointros,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),range2(A,B,R))) ) ).

% Range.intros
tff(fact_6642_RangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),range2(B,A,R)))
     => ~ ! [A5: B] : ~ pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A5),B2)),R)) ) ).

% RangeE
tff(fact_6643_Range__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,R: set(product_prod(B,A))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),range2(B,A,R)))
    <=> ? [Y5: B] : pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Y5),A3)),R)) ) ).

% Range_iff
tff(fact_6644_subset__Image1__Image1__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_preorder_on(A,field2(A,R),R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field2(A,R)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R,aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))),image(A,A,R,aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))))
          <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R)) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_6645_Rangep__Range__eq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),X5: B] :
      ( pp(aa(B,bool,rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X5))
    <=> pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X5),range2(A,B,R))) ) ).

% Rangep_Range_eq
tff(fact_6646_ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B)),A6: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),image(A,B,R,A6))) ) ) ).

% ImageI
tff(fact_6647_Image__singleton__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A)),A3: B] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),image(B,A,R,aa(set(B),set(B),insert(B,A3),bot_bot(set(B))))))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B2)),R)) ) ).

% Image_singleton_iff
tff(fact_6648_ImageE,axiom,
    ! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A)),A6: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),image(B,A,R,A6)))
     => ~ ! [X3: B] :
            ( pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),B2)),R))
           => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A6)) ) ) ).

% ImageE
tff(fact_6649_Image__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A)),A6: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),image(B,A,R,A6)))
    <=> ? [X4: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A6))
          & pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),B2)),R)) ) ) ).

% Image_iff
tff(fact_6650_rev__ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,A6: set(A),B2: B,R: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
     => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B2),image(A,B,R,A6))) ) ) ).

% rev_ImageI
tff(fact_6651_Image__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(A)] : image(A,B,R,S2) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_agg(set(product_prod(A,B)),fun(set(A),fun(B,bool)),R),S2)) ).

% Image_def
tff(fact_6652_Image__singleton,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),A3: B] : image(B,A,R,aa(set(B),set(B),insert(B,A3),bot_bot(set(B)))) = aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_agh(set(product_prod(B,A)),fun(B,fun(A,bool)),R),A3)) ).

% Image_singleton
tff(fact_6653_subset__Image__Image__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A6: set(A),B6: set(A)] :
      ( order_preorder_on(A,field2(A,R),R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),field2(A,R)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R,A6)),image(A,A,R,B6)))
          <=> ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
               => ? [Xa3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),B6))
                    & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X4)),R)) ) ) ) ) ) ) ).

% subset_Image_Image_iff
tff(fact_6654_Range__def,axiom,
    ! [B: $tType,A: $tType,X5: set(product_prod(A,B))] : range2(A,B,X5) = aa(fun(B,bool),set(B),collect(B),rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),X5))) ).

% Range_def
tff(fact_6655_less__eq__int__def,axiom,
    ord_less_eq(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(int,fun(int,bool)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(int,bool),rep_Integ,map_fun(int,product_prod(nat,nat),bool,bool,rep_Integ,id(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_kn(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).

% less_eq_int_def
tff(fact_6656_add_Ogroup__axioms,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => group(A,plus_plus(A),zero_zero(A),uminus_uminus(A)) ) ).

% add.group_axioms
tff(fact_6657_group_Oleft__cancel,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A,C3: A] :
      ( group(A,F3,Z2,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F3,A3),B2) = aa(A,A,aa(A,fun(A,A),F3,A3),C3) )
      <=> ( B2 = C3 ) ) ) ).

% group.left_cancel
tff(fact_6658_group_Oleft__inverse,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F3,Z2,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F3,aa(A,A,Inverse,A3)),A3) = Z2 ) ) ).

% group.left_inverse
tff(fact_6659_group_Oright__cancel,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),B2: A,A3: A,C3: A] :
      ( group(A,F3,Z2,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F3,B2),A3) = aa(A,A,aa(A,fun(A,A),F3,C3),A3) )
      <=> ( B2 = C3 ) ) ) ).

% group.right_cancel
tff(fact_6660_group_Oright__inverse,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F3,Z2,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F3,A3),aa(A,A,Inverse,A3)) = Z2 ) ) ).

% group.right_inverse
tff(fact_6661_group_Oinverse__unique,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A] :
      ( group(A,F3,Z2,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F3,A3),B2) = Z2 )
       => ( aa(A,A,Inverse,A3) = B2 ) ) ) ).

% group.inverse_unique
tff(fact_6662_group_Oinverse__inverse,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F3,Z2,Inverse)
     => ( aa(A,A,Inverse,aa(A,A,Inverse,A3)) = A3 ) ) ).

% group.inverse_inverse
tff(fact_6663_group_Oinverse__neutral,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group(A,F3,Z2,Inverse)
     => ( aa(A,A,Inverse,Z2) = Z2 ) ) ).

% group.inverse_neutral
tff(fact_6664_group_Ogroup__left__neutral,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F3,Z2,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F3,Z2),A3) = A3 ) ) ).

% group.group_left_neutral
tff(fact_6665_group_Oinverse__distrib__swap,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A] :
      ( group(A,F3,Z2,Inverse)
     => ( aa(A,A,Inverse,aa(A,A,aa(A,fun(A,A),F3,A3),B2)) = aa(A,A,aa(A,fun(A,A),F3,aa(A,A,Inverse,B2)),aa(A,A,Inverse,A3)) ) ) ).

% group.inverse_distrib_swap
tff(fact_6666_map__add__map__of__foldr,axiom,
    ! [B: $tType,A: $tType,M2: fun(A,option(B)),Ps: list(product_prod(A,B))] : map_add(A,B,M2,map_of(A,B,Ps)) = foldr(product_prod(A,B),fun(A,option(B)),aa(fun(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),fun(product_prod(A,B),fun(fun(A,option(B)),fun(A,option(B)))),product_case_prod(A,B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_agi(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))))),Ps,M2) ).

% map_add_map_of_foldr
tff(fact_6667_wo__rel_OisMinim__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A6: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( bNF_We4791949203932849705sMinim(A,R,A6,B2)
      <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A6))
          & ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
             => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),X4)),R)) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_6668_map__add__find__right,axiom,
    ! [B: $tType,A: $tType,N: fun(B,option(A)),K2: B,Xx: A,M2: fun(B,option(A))] :
      ( ( aa(B,option(A),N,K2) = aa(A,option(A),some(A),Xx) )
     => ( aa(B,option(A),map_add(B,A,M2,N),K2) = aa(A,option(A),some(A),Xx) ) ) ).

% map_add_find_right
tff(fact_6669_map__add__upd,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),G3: fun(A,option(B)),X: A,Y: B] : map_add(A,B,F3,fun_upd(A,option(B),G3,X,aa(B,option(B),some(B),Y))) = fun_upd(A,option(B),map_add(A,B,F3,G3),X,aa(B,option(B),some(B),Y)) ).

% map_add_upd
tff(fact_6670_map__add__SomeD,axiom,
    ! [B: $tType,A: $tType,M2: fun(B,option(A)),N: fun(B,option(A)),K2: B,X: A] :
      ( ( aa(B,option(A),map_add(B,A,M2,N),K2) = aa(A,option(A),some(A),X) )
     => ( ( aa(B,option(A),N,K2) = aa(A,option(A),some(A),X) )
        | ( ( aa(B,option(A),N,K2) = none(A) )
          & ( aa(B,option(A),M2,K2) = aa(A,option(A),some(A),X) ) ) ) ) ).

% map_add_SomeD
tff(fact_6671_map__add__Some__iff,axiom,
    ! [B: $tType,A: $tType,M2: fun(B,option(A)),N: fun(B,option(A)),K2: B,X: A] :
      ( ( aa(B,option(A),map_add(B,A,M2,N),K2) = aa(A,option(A),some(A),X) )
    <=> ( ( aa(B,option(A),N,K2) = aa(A,option(A),some(A),X) )
        | ( ( aa(B,option(A),N,K2) = none(A) )
          & ( aa(B,option(A),M2,K2) = aa(A,option(A),some(A),X) ) ) ) ) ).

% map_add_Some_iff
tff(fact_6672_map__add__def,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),X5: A] : aa(A,option(B),map_add(A,B,M1,M22),X5) = aa(option(B),option(B),aa(fun(B,option(B)),fun(option(B),option(B)),aa(option(B),fun(fun(B,option(B)),fun(option(B),option(B))),case_option(option(B),B),aa(A,option(B),M1,X5)),some(B)),aa(A,option(B),M22,X5)) ).

% map_add_def
tff(fact_6673_map__add__upd__left,axiom,
    ! [A: $tType,B: $tType,M2: A,E22: fun(A,option(B)),E1: fun(A,option(B)),U1: B] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M2),dom(A,B,E22)))
     => ( map_add(A,B,fun_upd(A,option(B),E1,M2,aa(B,option(B),some(B),U1)),E22) = fun_upd(A,option(B),map_add(A,B,E1,E22),M2,aa(B,option(B),some(B),U1)) ) ) ).

% map_add_upd_left
tff(fact_6674_iteratesp_Omono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A)] : pp(aa(fun(fun(A,bool),fun(A,bool)),bool,order_mono(fun(A,bool),fun(A,bool)),aTP_Lamp_agj(fun(A,A),fun(fun(A,bool),fun(A,bool)),F3))) ) ).

% iteratesp.mono
tff(fact_6675_inv__o__cancel,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( aa(fun(A,B),fun(A,A),comp(B,A,A,hilbert_inv_into(A,B,top_top(set(A)),F3)),F3) = id(A) ) ) ).

% inv_o_cancel
tff(fact_6676_inv__into__f__f,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),X: A] :
      ( inj_on(A,B,F3,A6)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
       => ( aa(B,A,hilbert_inv_into(A,B,A6,F3),aa(A,B,F3,X)) = X ) ) ) ).

% inv_into_f_f
tff(fact_6677_inv__identity,axiom,
    ! [A: $tType,X5: A] : aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),aTP_Lamp_abo(A,A)),X5) = X5 ).

% inv_identity
tff(fact_6678_inv__id,axiom,
    ! [A: $tType] : hilbert_inv_into(A,A,top_top(set(A)),id(A)) = id(A) ).

% inv_id
tff(fact_6679_inv__into__image__cancel,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),S3: set(A)] :
      ( inj_on(A,B,F3,A6)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S3),A6))
       => ( image2(B,A,hilbert_inv_into(A,B,A6,F3),image2(A,B,F3,S3)) = S3 ) ) ) ).

% inv_into_image_cancel
tff(fact_6680_o__inv__o__cancel,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,B),G3: fun(A,C)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( aa(fun(A,B),fun(A,C),comp(B,C,A,aa(fun(B,A),fun(B,C),comp(A,C,B,G3),hilbert_inv_into(A,B,top_top(set(A)),F3))),F3) = G3 ) ) ).

% o_inv_o_cancel
tff(fact_6681_image__inv__into__cancel,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A6: set(B),A10: set(A),B12: set(A)] :
      ( ( image2(B,A,F3,A6) = A10 )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B12),A10))
       => ( image2(B,A,F3,image2(A,B,hilbert_inv_into(B,A,A6,F3),B12)) = B12 ) ) ) ).

% image_inv_into_cancel
tff(fact_6682_chain__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: A] : comple1602240252501008431_chain(A,ord_less_eq(A),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) ) ).

% chain_singleton
tff(fact_6683_ccpo__Sup__upper,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A6: set(A),X: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A6)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A6))) ) ) ) ).

% ccpo_Sup_upper
tff(fact_6684_ccpo__Sup__least,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A6: set(A),Z2: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A6)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A6)),Z2)) ) ) ) ).

% ccpo_Sup_least
tff(fact_6685_bij__imp__bij__inv,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( bij_betw(A,B,F3,top_top(set(A)),top_top(set(B)))
     => bij_betw(B,A,hilbert_inv_into(A,B,top_top(set(A)),F3),top_top(set(B)),top_top(set(A))) ) ).

% bij_imp_bij_inv
tff(fact_6686_bij__inv__eq__iff,axiom,
    ! [A: $tType,B: $tType,P: fun(A,B),X: A,Y: B] :
      ( bij_betw(A,B,P,top_top(set(A)),top_top(set(B)))
     => ( ( X = aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),P),Y) )
      <=> ( aa(A,B,P,X) = Y ) ) ) ).

% bij_inv_eq_iff
tff(fact_6687_inv__inv__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( bij_betw(A,B,F3,top_top(set(A)),top_top(set(B)))
     => ( hilbert_inv_into(B,A,top_top(set(B)),hilbert_inv_into(A,B,top_top(set(A)),F3)) = F3 ) ) ).

% inv_inv_eq
tff(fact_6688_bij__betw__inv__into__right,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(A),A10: set(B),A4: B] :
      ( bij_betw(A,B,F3,A6,A10)
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A4),A10))
       => ( aa(A,B,F3,aa(B,A,hilbert_inv_into(A,B,A6,F3),A4)) = A4 ) ) ) ).

% bij_betw_inv_into_right
tff(fact_6689_bij__betw__inv__into__left,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),A10: set(B),A3: A] :
      ( bij_betw(A,B,F3,A6,A10)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
       => ( aa(B,A,hilbert_inv_into(A,B,A6,F3),aa(A,B,F3,A3)) = A3 ) ) ) ).

% bij_betw_inv_into_left
tff(fact_6690_inv__into__inv__into__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),A10: set(B),A3: A] :
      ( bij_betw(A,B,F3,A6,A10)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),A6))
       => ( aa(A,B,hilbert_inv_into(B,A,A10,hilbert_inv_into(A,B,A6,F3)),A3) = aa(A,B,F3,A3) ) ) ) ).

% inv_into_inv_into_eq
tff(fact_6691_bij__betw__inv__into,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),B6: set(B)] :
      ( bij_betw(A,B,F3,A6,B6)
     => bij_betw(B,A,hilbert_inv_into(A,B,A6,F3),B6,A6) ) ).

% bij_betw_inv_into
tff(fact_6692_inv__equality,axiom,
    ! [A: $tType,B: $tType,G3: fun(B,A),F3: fun(A,B)] :
      ( ! [X3: A] : aa(B,A,G3,aa(A,B,F3,X3)) = X3
     => ( ! [Y3: B] : aa(A,B,F3,aa(B,A,G3,Y3)) = Y3
       => ( hilbert_inv_into(A,B,top_top(set(A)),F3) = G3 ) ) ) ).

% inv_equality
tff(fact_6693_inj__imp__inv__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),G3: fun(B,A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( ! [X3: B] : aa(A,B,F3,aa(B,A,G3,X3)) = X3
       => ( hilbert_inv_into(A,B,top_top(set(A)),F3) = G3 ) ) ) ).

% inj_imp_inv_eq
tff(fact_6694_inv__f__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X: A,Y: B] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( ( aa(A,B,F3,X) = Y )
       => ( aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F3),Y) = X ) ) ) ).

% inv_f_eq
tff(fact_6695_inv__f__f,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X: A] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F3),aa(A,B,F3,X)) = X ) ) ).

% inv_f_f
tff(fact_6696_inv__into__f__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),X: A,Y: B] :
      ( inj_on(A,B,F3,A6)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
       => ( ( aa(A,B,F3,X) = Y )
         => ( aa(B,A,hilbert_inv_into(A,B,A6,F3),Y) = X ) ) ) ) ).

% inv_into_f_eq
tff(fact_6697_f__inv__into__f,axiom,
    ! [B: $tType,A: $tType,Y: A,F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),image2(B,A,F3,A6)))
     => ( aa(B,A,F3,aa(A,B,hilbert_inv_into(B,A,A6,F3),Y)) = Y ) ) ).

% f_inv_into_f
tff(fact_6698_inv__into__into,axiom,
    ! [A: $tType,B: $tType,X: A,F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),image2(B,A,F3,A6)))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,hilbert_inv_into(B,A,A6,F3),X)),A6)) ) ).

% inv_into_into
tff(fact_6699_inv__into__injective,axiom,
    ! [A: $tType,B: $tType,A6: set(A),F3: fun(A,B),X: B,Y: B] :
      ( ( aa(B,A,hilbert_inv_into(A,B,A6,F3),X) = aa(B,A,hilbert_inv_into(A,B,A6,F3),Y) )
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),image2(A,B,F3,A6)))
       => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),image2(A,B,F3,A6)))
         => ( X = Y ) ) ) ) ).

% inv_into_injective
tff(fact_6700_surj__imp__inv__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),G3: fun(A,B)] :
      ( ( image2(B,A,F3,top_top(set(B))) = top_top(set(A)) )
     => ( ! [X3: B] : aa(A,B,G3,aa(B,A,F3,X3)) = X3
       => ( hilbert_inv_into(B,A,top_top(set(B)),F3) = G3 ) ) ) ).

% surj_imp_inv_eq
tff(fact_6701_image__f__inv__f,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A6: set(A)] :
      ( ( image2(B,A,F3,top_top(set(B))) = top_top(set(A)) )
     => ( image2(B,A,F3,image2(A,B,hilbert_inv_into(B,A,top_top(set(B)),F3),A6)) = A6 ) ) ).

% image_f_inv_f
tff(fact_6702_surj__iff__all,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A)] :
      ( ( image2(B,A,F3,top_top(set(B))) = top_top(set(A)) )
    <=> ! [X4: A] : aa(B,A,F3,aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F3),X4)) = X4 ) ).

% surj_iff_all
tff(fact_6703_surj__f__inv__f,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Y: A] :
      ( ( image2(B,A,F3,top_top(set(B))) = top_top(set(A)) )
     => ( aa(B,A,F3,aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F3),Y)) = Y ) ) ).

% surj_f_inv_f
tff(fact_6704_mono__compose,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Q: fun(A,fun(B,C)),F3: fun(D,B)] :
          ( pp(aa(fun(A,fun(B,C)),bool,order_mono(A,fun(B,C)),Q))
         => pp(aa(fun(A,fun(D,C)),bool,order_mono(A,fun(D,C)),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_agk(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F3))) ) ) ).

% mono_compose
tff(fact_6705_inv__into__def2,axiom,
    ! [A: $tType,B: $tType,A6: set(A),F3: fun(A,B),X: B] : aa(B,A,hilbert_inv_into(A,B,A6,F3),X) = fChoice(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_lf(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A6),F3),X)) ).

% inv_into_def2
tff(fact_6706_inv__into__def,axiom,
    ! [A: $tType,B: $tType,A6: set(A),F3: fun(A,B),X5: B] : aa(B,A,hilbert_inv_into(A,B,A6,F3),X5) = fChoice(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_lf(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A6),F3),X5)) ).

% inv_into_def
tff(fact_6707_inv__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),X5: A] : aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F3),X5) = fChoice(B,aa(A,fun(B,bool),aTP_Lamp_agl(fun(B,A),fun(A,fun(B,bool)),F3),X5)) ).

% inv_def
tff(fact_6708_inj__transfer,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),P2: fun(A,bool),X: A] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( ! [Y3: B] :
            ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y3),image2(A,B,F3,top_top(set(A)))))
           => pp(aa(A,bool,P2,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F3),Y3))) )
       => pp(aa(A,bool,P2,X)) ) ) ).

% inj_transfer
tff(fact_6709_image__inv__f__f,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( image2(B,A,hilbert_inv_into(A,B,top_top(set(A)),F3),image2(A,B,F3,A6)) = A6 ) ) ).

% image_inv_f_f
tff(fact_6710_inj__imp__surj__inv,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( image2(B,A,hilbert_inv_into(A,B,top_top(set(A)),F3),top_top(set(B))) = top_top(set(A)) ) ) ).

% inj_imp_surj_inv
tff(fact_6711_surj__imp__inj__inv,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A)] :
      ( ( image2(B,A,F3,top_top(set(B))) = top_top(set(A)) )
     => inj_on(A,B,hilbert_inv_into(B,A,top_top(set(B)),F3),top_top(set(A))) ) ).

% surj_imp_inj_inv
tff(fact_6712_inj__on__inv__into,axiom,
    ! [B: $tType,A: $tType,B6: set(A),F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),image2(B,A,F3,A6)))
     => inj_on(A,B,hilbert_inv_into(B,A,A6,F3),B6) ) ).

% inj_on_inv_into
tff(fact_6713_inv__into__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,B),G3: fun(C,A),A6: set(C),X: B] :
      ( inj_on(A,B,F3,image2(C,A,G3,A6))
     => ( inj_on(C,A,G3,A6)
       => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),image2(A,B,F3,image2(C,A,G3,A6))))
         => ( aa(B,C,hilbert_inv_into(C,B,A6,aa(fun(C,A),fun(C,B),comp(A,B,C,F3),G3)),X) = aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,hilbert_inv_into(C,A,A6,G3)),hilbert_inv_into(A,B,image2(C,A,G3,A6),F3)),X) ) ) ) ) ).

% inv_into_comp
tff(fact_6714_bij__betw__inv__into__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),A10: set(B),B6: set(A),B12: set(B)] :
      ( bij_betw(A,B,F3,A6,A10)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),A6))
       => ( ( image2(A,B,F3,B6) = B12 )
         => bij_betw(B,A,hilbert_inv_into(A,B,A6,F3),B12,B6) ) ) ) ).

% bij_betw_inv_into_subset
tff(fact_6715_inv__unique__comp,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),G3: fun(A,B)] :
      ( ( aa(fun(A,B),fun(A,A),comp(B,A,A,F3),G3) = id(A) )
     => ( ( aa(fun(B,A),fun(B,B),comp(A,B,B,G3),F3) = id(B) )
       => ( hilbert_inv_into(B,A,top_top(set(B)),F3) = G3 ) ) ) ).

% inv_unique_comp
tff(fact_6716_o__inv__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,B),G3: fun(C,A)] :
      ( bij_betw(A,B,F3,top_top(set(A)),top_top(set(B)))
     => ( bij_betw(C,A,G3,top_top(set(C)),top_top(set(A)))
       => ( hilbert_inv_into(C,B,top_top(set(C)),aa(fun(C,A),fun(C,B),comp(A,B,C,F3),G3)) = aa(fun(B,A),fun(B,C),comp(A,C,B,hilbert_inv_into(C,A,top_top(set(C)),G3)),hilbert_inv_into(A,B,top_top(set(A)),F3)) ) ) ) ).

% o_inv_distrib
tff(fact_6717_mono__inv,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,order_mono(A,B),F3))
         => ( bij_betw(A,B,F3,top_top(set(A)),top_top(set(B)))
           => pp(aa(fun(B,A),bool,order_mono(B,A),hilbert_inv_into(A,B,top_top(set(A)),F3))) ) ) ) ).

% mono_inv
tff(fact_6718_inv__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat] :
      ( bij_betw(A,A,F3,top_top(set(A)),top_top(set(A)))
     => ( hilbert_inv_into(A,A,top_top(set(A)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),hilbert_inv_into(A,A,top_top(set(A)),F3)) ) ) ).

% inv_fn
tff(fact_6719_bij__image__Collect__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),P2: fun(A,bool)] :
      ( bij_betw(A,B,F3,top_top(set(A)),top_top(set(B)))
     => ( image2(A,B,F3,aa(fun(A,bool),set(A),collect(A),P2)) = aa(fun(B,bool),set(B),collect(B),aa(fun(A,bool),fun(B,bool),aTP_Lamp_agm(fun(A,B),fun(fun(A,bool),fun(B,bool)),F3),P2)) ) ) ).

% bij_image_Collect_eq
tff(fact_6720_surj__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A)] :
      ( ( image2(B,A,F3,top_top(set(B))) = top_top(set(A)) )
    <=> ( aa(fun(A,B),fun(A,A),comp(B,A,A,F3),hilbert_inv_into(B,A,top_top(set(B)),F3)) = id(A) ) ) ).

% surj_iff
tff(fact_6721_inj__imp__bij__betw__inv,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),M5: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => bij_betw(B,A,hilbert_inv_into(A,B,top_top(set(A)),F3),image2(A,B,F3,M5),M5) ) ).

% inj_imp_bij_betw_inv
tff(fact_6722_inj__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
    <=> ( aa(fun(A,B),fun(A,A),comp(B,A,A,hilbert_inv_into(A,B,top_top(set(A)),F3)),F3) = id(A) ) ) ).

% inj_iff
tff(fact_6723_bij__vimage__eq__inv__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A6: set(B)] :
      ( bij_betw(A,B,F3,top_top(set(A)),top_top(set(B)))
     => ( vimage(A,B,F3,A6) = image2(B,A,hilbert_inv_into(A,B,top_top(set(A)),F3),A6) ) ) ).

% bij_vimage_eq_inv_image
tff(fact_6724_fn__o__inv__fn__is__id,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat] :
      ( bij_betw(A,A,F3,top_top(set(A)),top_top(set(A)))
     => ! [X5: A] : aa(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),hilbert_inv_into(A,A,top_top(set(A)),F3))),X5) = X5 ) ).

% fn_o_inv_fn_is_id
tff(fact_6725_inv__fn__o__fn__is__id,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat] :
      ( bij_betw(A,A,F3,top_top(set(A)),top_top(set(A)))
     => ! [X5: A] : aa(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),hilbert_inv_into(A,A,top_top(set(A)),F3))),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)),X5) = X5 ) ).

% inv_fn_o_fn_is_id
tff(fact_6726_in__chain__finite,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A6: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A6)
         => ( pp(aa(set(A),bool,finite_finite2(A),A6))
           => ( ( A6 != bot_bot(set(A)) )
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,complete_Sup_Sup(A),A6)),A6)) ) ) ) ) ).

% in_chain_finite
tff(fact_6727_iteratesp__def,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X5: fun(A,A)] : comple7512665784863727008ratesp(A,X5) = complete_lattice_lfp(fun(A,bool),aTP_Lamp_agj(fun(A,A),fun(fun(A,bool),fun(A,bool)),X5)) ) ).

% iteratesp_def
tff(fact_6728_iteratesp_OSup,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [M5: set(A),F3: fun(A,A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),M5)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),M5))
               => pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X3)) )
           => pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),aa(set(A),A,complete_Sup_Sup(A),M5))) ) ) ) ).

% iteratesp.Sup
tff(fact_6729_iteratesp_Ocases,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A),A3: A] :
          ( pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),A3))
         => ( ! [X3: A] :
                ( ( A3 = aa(A,A,F3,X3) )
               => ~ pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X3)) )
           => ~ ! [M8: set(A)] :
                  ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M8) )
                 => ( comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                   => ~ ! [X5: A] :
                          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),M8))
                         => pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X5)) ) ) ) ) ) ) ).

% iteratesp.cases
tff(fact_6730_iteratesp_Osimps,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A),A3: A] :
          ( pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),A3))
        <=> ( ? [X4: A] :
                ( ( A3 = aa(A,A,F3,X4) )
                & pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X4)) )
            | ? [M9: set(A)] :
                ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M9) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M9)
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),M9))
                   => pp(aa(A,bool,comple7512665784863727008ratesp(A,F3),X4)) ) ) ) ) ) ).

% iteratesp.simps
tff(fact_6731_strict__mono__inv__on__range,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_strict_mono(A,B,F3)
         => strict_mono_on(B,A,hilbert_inv_into(A,B,top_top(set(A)),F3),image2(A,B,F3,top_top(set(A)))) ) ) ).

% strict_mono_inv_on_range
tff(fact_6732_bijection_Oinv__comp__left,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( hilbert_bijection(A,F3)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,hilbert_inv_into(A,A,top_top(set(A)),F3)),F3) = id(A) ) ) ).

% bijection.inv_comp_left
tff(fact_6733_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F3: fun(A,B),A6: set(A),X: A,Y: A] :
          ( strict_mono_on(A,B,F3,A6)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A6))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_6734_bijection_OeqI,axiom,
    ! [A: $tType,F3: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F3)
     => ( ( aa(A,A,F3,A3) = aa(A,A,F3,B2) )
       => ( A3 = B2 ) ) ) ).

% bijection.eqI
tff(fact_6735_bijection_Oeq__iff,axiom,
    ! [A: $tType,F3: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F3)
     => ( ( aa(A,A,F3,A3) = aa(A,A,F3,B2) )
      <=> ( A3 = B2 ) ) ) ).

% bijection.eq_iff
tff(fact_6736_bijection_Osurj,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( hilbert_bijection(A,F3)
     => ( image2(A,A,F3,top_top(set(A))) = top_top(set(A)) ) ) ).

% bijection.surj
tff(fact_6737_bijection_Oinj,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( hilbert_bijection(A,F3)
     => inj_on(A,A,F3,top_top(set(A))) ) ).

% bijection.inj
tff(fact_6738_bijection__def,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( hilbert_bijection(A,F3)
    <=> bij_betw(A,A,F3,top_top(set(A)),top_top(set(A))) ) ).

% bijection_def
tff(fact_6739_bijection_Ointro,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( bij_betw(A,A,F3,top_top(set(A)),top_top(set(A)))
     => hilbert_bijection(A,F3) ) ).

% bijection.intro
tff(fact_6740_bijection_Obij,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( hilbert_bijection(A,F3)
     => bij_betw(A,A,F3,top_top(set(A)),top_top(set(A))) ) ).

% bijection.bij
tff(fact_6741_bijection_Oinv__right__eq__iff,axiom,
    ! [A: $tType,F3: fun(A,A),B2: A,A3: A] :
      ( hilbert_bijection(A,F3)
     => ( ( B2 = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F3),A3) )
      <=> ( aa(A,A,F3,B2) = A3 ) ) ) ).

% bijection.inv_right_eq_iff
tff(fact_6742_bijection_Oinv__left__eq__iff,axiom,
    ! [A: $tType,F3: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F3)
     => ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F3),A3) = B2 )
      <=> ( aa(A,A,F3,B2) = A3 ) ) ) ).

% bijection.inv_left_eq_iff
tff(fact_6743_bijection_Oeq__inv__iff,axiom,
    ! [A: $tType,F3: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F3)
     => ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F3),A3) = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F3),B2) )
      <=> ( A3 = B2 ) ) ) ).

% bijection.eq_inv_iff
tff(fact_6744_bijection_Oinv__right,axiom,
    ! [A: $tType,F3: fun(A,A),A3: A] :
      ( hilbert_bijection(A,F3)
     => ( aa(A,A,F3,aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F3),A3)) = A3 ) ) ).

% bijection.inv_right
tff(fact_6745_bijection_Oinv__left,axiom,
    ! [A: $tType,F3: fun(A,A),A3: A] :
      ( hilbert_bijection(A,F3)
     => ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F3),aa(A,A,F3,A3)) = A3 ) ) ).

% bijection.inv_left
tff(fact_6746_bijection_Oeq__invI,axiom,
    ! [A: $tType,F3: fun(A,A),A3: A,B2: A] :
      ( hilbert_bijection(A,F3)
     => ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F3),A3) = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F3),B2) )
       => ( A3 = B2 ) ) ) ).

% bijection.eq_invI
tff(fact_6747_bijection_Osurj__inv,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( hilbert_bijection(A,F3)
     => ( image2(A,A,hilbert_inv_into(A,A,top_top(set(A)),F3),top_top(set(A))) = top_top(set(A)) ) ) ).

% bijection.surj_inv
tff(fact_6748_bijection_Oinj__inv,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( hilbert_bijection(A,F3)
     => inj_on(A,A,hilbert_inv_into(A,A,top_top(set(A)),F3),top_top(set(A))) ) ).

% bijection.inj_inv
tff(fact_6749_bijection_Obij__inv,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( hilbert_bijection(A,F3)
     => bij_betw(A,A,hilbert_inv_into(A,A,top_top(set(A)),F3),top_top(set(A)),top_top(set(A))) ) ).

% bijection.bij_inv
tff(fact_6750_bijection_Oinv__comp__right,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( hilbert_bijection(A,F3)
     => ( aa(fun(A,A),fun(A,A),comp(A,A,A,F3),hilbert_inv_into(A,A,top_top(set(A)),F3)) = id(A) ) ) ).

% bijection.inv_comp_right
tff(fact_6751_admissible__chfin,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P2: fun(A,bool)] :
          ( ! [S4: set(A)] :
              ( comple1602240252501008431_chain(A,ord_less_eq(A),S4)
             => pp(aa(set(A),bool,finite_finite2(A),S4)) )
         => comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P2) ) ) ).

% admissible_chfin
tff(fact_6752_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [A: $tType,X: fun(A,fun(A,A)),Xa2: option(A),Xb: option(A),Y: option(A)] :
      ( ( aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,X),Xa2),Xb) = Y )
     => ( accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),Xa2),Xb)))
       => ( ( ( Xa2 = none(A) )
           => ( ( Y = none(A) )
             => ~ accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Xb))) ) )
         => ( ! [V2: A] :
                ( ( Xa2 = aa(A,option(A),some(A),V2) )
               => ( ( Xb = none(A) )
                 => ( ( Y = none(A) )
                   => ~ accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V2)),none(A)))) ) ) )
           => ~ ! [A5: A] :
                  ( ( Xa2 = aa(A,option(A),some(A),A5) )
                 => ! [B4: A] :
                      ( ( Xb = aa(A,option(A),some(A),B4) )
                     => ( ( Y = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),X,A5),B4)) )
                       => ~ accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),A5)),aa(A,option(A),some(A),B4)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_6753_admissible__disj,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P2: fun(A,bool),Q: fun(A,bool)] :
          ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P2)
         => ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),Q)
           => comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_agn(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P2),Q)) ) ) ) ).

% admissible_disj
tff(fact_6754_Domain__insert,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] : domain(A,B,aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R)) = aa(set(A),set(A),insert(A,A3),domain(A,B,R)) ).

% Domain_insert
tff(fact_6755_ndepth__Push__Node__aux,axiom,
    ! [A: $tType,I2: nat,F3: fun(nat,sum_sum(A,nat)),K2: nat] :
      ( ( case_nat(sum_sum(A,nat),aa(nat,sum_sum(A,nat),sum_Inr(nat,A),aa(nat,nat,suc,I2)),F3,K2) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_ago(fun(nat,sum_sum(A,nat)),fun(nat,bool),F3)))),K2)) ) ).

% ndepth_Push_Node_aux
tff(fact_6756_Domain_Ocases,axiom,
    ! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),domain(A,B,R)))
     => ~ ! [B4: B] : ~ pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B4)),R)) ) ).

% Domain.cases
tff(fact_6757_Domain_Osimps,axiom,
    ! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),domain(A,B,R)))
    <=> ? [A7: A,B5: B] :
          ( ( A3 = A7 )
          & pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5)),R)) ) ) ).

% Domain.simps
tff(fact_6758_Domain_ODomainI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),domain(A,B,R))) ) ).

% Domain.DomainI
tff(fact_6759_DomainE,axiom,
    ! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),domain(A,B,R)))
     => ~ ! [B4: B] : ~ pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B4)),R)) ) ).

% DomainE
tff(fact_6760_Domain__iff,axiom,
    ! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),domain(A,B,R)))
    <=> ? [Y5: B] : pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),Y5)),R)) ) ).

% Domain_iff
tff(fact_6761_Not__Domain__rtrancl,axiom,
    ! [A: $tType,X: A,R2: set(product_prod(A,A)),Y: A] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),domain(A,A,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R2)))
      <=> ( X = Y ) ) ) ).

% Not_Domain_rtrancl
tff(fact_6762_Domain__unfold,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] : domain(A,B,R) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_agp(set(product_prod(A,B)),fun(A,bool),R)) ).

% Domain_unfold
tff(fact_6763_sum_Osize_I4_J,axiom,
    ! [B: $tType,A: $tType,X2: B] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),aa(B,sum_sum(A,B),sum_Inr(B,A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ).

% sum.size(4)
tff(fact_6764_sum_Osize__gen_I2_J,axiom,
    ! [A: $tType,B: $tType,Xa2: fun(A,nat),X: fun(B,nat),X2: B] : basic_BNF_size_sum(A,B,Xa2,X,aa(B,sum_sum(A,B),sum_Inr(B,A),X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,X,X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% sum.size_gen(2)
tff(fact_6765_Node__def,axiom,
    ! [A: $tType,B: $tType] : old_Node(B,A) = aa(fun(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool),set(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),collect(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),aTP_Lamp_agq(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool)) ).

% Node_def
tff(fact_6766_Node__K0__I,axiom,
    ! [B: $tType,A: $tType,A3: sum_sum(B,nat)] : pp(aa(set(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),bool,aa(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),fun(set(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),bool),member(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),aa(sum_sum(B,nat),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aa(fun(nat,sum_sum(A,nat)),fun(sum_sum(B,nat),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),product_Pair(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aTP_Lamp_agr(nat,sum_sum(A,nat))),A3)),old_Node(A,B))) ).

% Node_K0_I
tff(fact_6767_Push__neq__K0,axiom,
    ! [A: $tType,K2: nat,F3: fun(nat,sum_sum(A,nat))] :
      ~ ! [X3: nat] : old_Push(A,aa(nat,sum_sum(A,nat),sum_Inr(nat,A),aa(nat,nat,suc,K2)),F3,X3) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ).

% Push_neq_K0
tff(fact_6768_ndepth__def,axiom,
    ! [B: $tType,A: $tType,N: old_node(A,B)] : old_ndepth(A,B,N) = aa(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),nat,aa(fun(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat)),fun(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),nat),product_case_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat),nat),aTP_Lamp_agt(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat))),old_Rep_Node(A,B,N)) ).

% ndepth_def
tff(fact_6769_ndepth__K0,axiom,
    ! [A: $tType,B: $tType,X: sum_sum(A,nat)] : old_ndepth(A,B,old_Abs_Node(B,A,aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aTP_Lamp_agu(nat,sum_sum(B,nat))),X))) = zero_zero(nat) ).

% ndepth_K0
tff(fact_6770_ndepth__Push__Node,axiom,
    ! [B: $tType,A: $tType,I2: nat,N: old_node(A,B)] : old_ndepth(A,B,aa(old_node(A,B),old_node(A,B),old_Push_Node(B,A,aa(nat,sum_sum(B,nat),sum_Inr(nat,B),aa(nat,nat,suc,I2))),N)) = aa(nat,nat,suc,old_ndepth(A,B,N)) ).

% ndepth_Push_Node
tff(fact_6771_Atom__def,axiom,
    ! [B: $tType,A: $tType,X5: sum_sum(A,nat)] : old_Atom(A,B,X5) = aa(set(old_node(A,B)),set(old_node(A,B)),insert(old_node(A,B),old_Abs_Node(B,A,aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aTP_Lamp_agu(nat,sum_sum(B,nat))),X5))),bot_bot(set(old_node(A,B)))) ).

% Atom_def
tff(fact_6772_Scons__def,axiom,
    ! [B: $tType,A: $tType,M5: set(old_node(A,B)),N6: set(old_node(A,B))] : old_Scons(A,B,M5,N6) = aa(set(old_node(A,B)),set(old_node(A,B)),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(old_node(A,B))),sup_sup(set(old_node(A,B))),image2(old_node(A,B),old_node(A,B),old_Push_Node(B,A,aa(nat,sum_sum(B,nat),sum_Inr(nat,B),one_one(nat))),M5)),image2(old_node(A,B),old_node(A,B),old_Push_Node(B,A,aa(nat,sum_sum(B,nat),sum_Inr(nat,B),aa(nat,nat,suc,one_one(nat)))),N6)) ).

% Scons_def
tff(fact_6773_ntrunc__0,axiom,
    ! [B: $tType,A: $tType,M5: set(old_node(A,B))] : old_ntrunc(A,B,zero_zero(nat),M5) = bot_bot(set(old_node(A,B))) ).

% ntrunc_0
tff(fact_6774_int__encode__def,axiom,
    ! [I2: int] : aa(int,nat,nat_int_encode,I2) = aa(sum_sum(nat,nat),nat,nat_sum_encode,if(sum_sum(nat,nat),aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I2),aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),nat2(I2)),aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),nat2(aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),I2)),one_one(int)))))) ).

% int_encode_def
tff(fact_6775_ntrunc__Scons,axiom,
    ! [B: $tType,A: $tType,K2: nat,M5: set(old_node(A,B)),N6: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,K2),old_Scons(A,B,M5,N6)) = old_Scons(A,B,old_ntrunc(A,B,K2,M5),old_ntrunc(A,B,K2,N6)) ).

% ntrunc_Scons
tff(fact_6776_ntrunc__Atom,axiom,
    ! [B: $tType,A: $tType,K2: nat,A3: sum_sum(A,nat)] : old_ntrunc(A,B,aa(nat,nat,suc,K2),old_Atom(A,B,A3)) = old_Atom(A,B,A3) ).

% ntrunc_Atom
tff(fact_6777_surj__sum__encode,axiom,
    image2(sum_sum(nat,nat),nat,nat_sum_encode,top_top(set(sum_sum(nat,nat)))) = top_top(set(nat)) ).

% surj_sum_encode
tff(fact_6778_inj__sum__encode,axiom,
    ! [A6: set(sum_sum(nat,nat))] : inj_on(sum_sum(nat,nat),nat,nat_sum_encode,A6) ).

% inj_sum_encode
tff(fact_6779_sum__encode__eq,axiom,
    ! [X: sum_sum(nat,nat),Y: sum_sum(nat,nat)] :
      ( ( aa(sum_sum(nat,nat),nat,nat_sum_encode,X) = aa(sum_sum(nat,nat),nat,nat_sum_encode,Y) )
    <=> ( X = Y ) ) ).

% sum_encode_eq
tff(fact_6780_bij__sum__encode,axiom,
    bij_betw(sum_sum(nat,nat),nat,nat_sum_encode,top_top(set(sum_sum(nat,nat))),top_top(set(nat))) ).

% bij_sum_encode
tff(fact_6781_le__sum__encode__Inl,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(sum_sum(nat,nat),nat,nat_sum_encode,aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),Y)))) ) ).

% le_sum_encode_Inl
tff(fact_6782_le__sum__encode__Inr,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(sum_sum(nat,nat),nat,nat_sum_encode,aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Y)))) ) ).

% le_sum_encode_Inr
tff(fact_6783_sum_Osize_I3_J,axiom,
    ! [A: $tType,B: $tType,X1: A] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),aa(A,sum_sum(A,B),sum_Inl(A,B),X1)) = aa(nat,nat,suc,zero_zero(nat)) ).

% sum.size(3)
tff(fact_6784_sum_Osize__gen_I1_J,axiom,
    ! [B: $tType,A: $tType,Xa2: fun(A,nat),X: fun(B,nat),X1: A] : basic_BNF_size_sum(A,B,Xa2,X,aa(A,sum_sum(A,B),sum_Inl(A,B),X1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa2,X1)),aa(nat,nat,suc,zero_zero(nat))) ).

% sum.size_gen(1)
tff(fact_6785_sum__decode__def,axiom,
    ! [N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% sum_decode_def
tff(fact_6786_ntrunc__one__In0,axiom,
    ! [B: $tType,A: $tType,M5: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,zero_zero(nat)),old_In0(A,B,M5)) = bot_bot(set(old_node(A,B))) ).

% ntrunc_one_In0
tff(fact_6787_sum__encode__inverse,axiom,
    ! [X: sum_sum(nat,nat)] : aa(nat,sum_sum(nat,nat),nat_sum_decode,aa(sum_sum(nat,nat),nat,nat_sum_encode,X)) = X ).

% sum_encode_inverse
tff(fact_6788_sum__decode__inverse,axiom,
    ! [N: nat] : aa(sum_sum(nat,nat),nat,nat_sum_encode,aa(nat,sum_sum(nat,nat),nat_sum_decode,N)) = N ).

% sum_decode_inverse
tff(fact_6789_ntrunc__In0,axiom,
    ! [B: $tType,A: $tType,K2: nat,M5: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,aa(nat,nat,suc,K2)),old_In0(A,B,M5)) = old_In0(A,B,old_ntrunc(A,B,aa(nat,nat,suc,K2),M5)) ).

% ntrunc_In0
tff(fact_6790_bij__sum__decode,axiom,
    bij_betw(nat,sum_sum(nat,nat),nat_sum_decode,top_top(set(nat)),top_top(set(sum_sum(nat,nat)))) ).

% bij_sum_decode
tff(fact_6791_sum__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,X) = aa(nat,sum_sum(nat,nat),nat_sum_decode,Y) )
    <=> ( X = Y ) ) ).

% sum_decode_eq
tff(fact_6792_inj__sum__decode,axiom,
    ! [A6: set(nat)] : inj_on(nat,sum_sum(nat,nat),nat_sum_decode,A6) ).

% inj_sum_decode
tff(fact_6793_surj__sum__decode,axiom,
    image2(nat,sum_sum(nat,nat),nat_sum_decode,top_top(set(nat))) = top_top(set(sum_sum(nat,nat))) ).

% surj_sum_decode
tff(fact_6794_nth__item_Opinduct,axiom,
    ! [A0: nat,P2: fun(nat,bool)] :
      ( accp(nat,nth_item_rel,A0)
     => ( ( accp(nat,nth_item_rel,zero_zero(nat))
         => pp(aa(nat,bool,P2,zero_zero(nat))) )
       => ( ! [N3: nat] :
              ( accp(nat,nth_item_rel,aa(nat,nat,suc,N3))
             => ( ! [A8: nat,Aa3: nat] :
                    ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N3) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),A8) )
                   => ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,A8) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),Aa3) )
                     => pp(aa(nat,bool,P2,Aa3)) ) )
               => ( ! [A8: nat,B7: nat] :
                      ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N3) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),A8) )
                     => ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,A8) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B7) )
                       => pp(aa(nat,bool,P2,B7)) ) )
                 => ( ! [B7: nat,Ba2: nat,X5: nat,Y4: nat] :
                        ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N3) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B7) )
                       => ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,B7) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Ba2) )
                         => ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X5),Y4) = aa(nat,product_prod(nat,nat),nat_prod_decode,Ba2) )
                           => pp(aa(nat,bool,P2,X5)) ) ) )
                   => ( ! [B7: nat,Ba2: nat,X5: nat,Y4: nat] :
                          ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N3) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B7) )
                         => ( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,B7) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Ba2) )
                           => ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X5),Y4) = aa(nat,product_prod(nat,nat),nat_prod_decode,Ba2) )
                             => pp(aa(nat,bool,P2,Y4)) ) ) )
                     => pp(aa(nat,bool,P2,aa(nat,nat,suc,N3))) ) ) ) ) )
         => pp(aa(nat,bool,P2,A0)) ) ) ) ).

% nth_item.pinduct
tff(fact_6795_ntrunc__one__In1,axiom,
    ! [B: $tType,A: $tType,M5: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,zero_zero(nat)),old_In1(A,B,M5)) = bot_bot(set(old_node(A,B))) ).

% ntrunc_one_In1
tff(fact_6796_ntrunc__In1,axiom,
    ! [B: $tType,A: $tType,K2: nat,M5: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,aa(nat,nat,suc,K2)),old_In1(A,B,M5)) = old_In1(A,B,old_ntrunc(A,B,aa(nat,nat,suc,K2),M5)) ).

% ntrunc_In1
tff(fact_6797_int__decode__def,axiom,
    ! [N: nat] : aa(nat,int,nat_int_decode,N) = aa(sum_sum(nat,nat),int,sum_case_sum(nat,int,nat,semiring_1_of_nat(int),aTP_Lamp_agv(nat,int)),aa(nat,sum_sum(nat,nat),nat_sum_decode,N)) ).

% int_decode_def
tff(fact_6798_sum__encode__def,axiom,
    ! [X: sum_sum(nat,nat)] : aa(sum_sum(nat,nat),nat,nat_sum_encode,X) = aa(sum_sum(nat,nat),nat,sum_case_sum(nat,nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aTP_Lamp_agw(nat,nat)),X) ).

% sum_encode_def
tff(fact_6799_ntrunc__Leaf,axiom,
    ! [B: $tType,A: $tType,K2: nat,A3: A] : old_ntrunc(A,B,aa(nat,nat,suc,K2),old_Leaf(A,B,A3)) = old_Leaf(A,B,A3) ).

% ntrunc_Leaf
tff(fact_6800_In0__def,axiom,
    ! [B: $tType,A: $tType,M5: set(old_node(A,B))] : old_In0(A,B,M5) = old_Scons(A,B,old_Numb(A,B,zero_zero(nat)),M5) ).

% In0_def
tff(fact_6801_ntrunc__Numb,axiom,
    ! [A: $tType,B: $tType,K2: nat,I2: nat] : old_ntrunc(A,B,aa(nat,nat,suc,K2),old_Numb(A,B,I2)) = old_Numb(A,B,I2) ).

% ntrunc_Numb
tff(fact_6802_prod_OPlus,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [A6: set(B),B6: set(C),G3: fun(sum_sum(B,C),A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A6))
         => ( pp(aa(set(C),bool,finite_finite2(C),B6))
           => ( aa(set(sum_sum(B,C)),A,aa(fun(sum_sum(B,C),A),fun(set(sum_sum(B,C)),A),groups7121269368397514597t_prod(sum_sum(B,C),A),G3),sum_Plus(B,C,A6,B6)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,sum_sum(B,C)),fun(B,A),comp(sum_sum(B,C),A,B,G3),sum_Inl(B,C))),A6)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(C,sum_sum(B,C)),fun(C,A),comp(sum_sum(B,C),A,C,G3),sum_Inr(C,B))),B6)) ) ) ) ) ).

% prod.Plus
tff(fact_6803_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),X: A,Y: A,Zs: list(A)] : arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))),X,arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) ) ).

% arg_min_list.simps(2)
tff(fact_6804_card__Plus__conv__if,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
      ( ( ( pp(aa(set(A),bool,finite_finite2(A),A6))
          & pp(aa(set(B),bool,finite_finite2(B),B6)) )
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A6,B6)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B6)) ) )
      & ( ~ ( pp(aa(set(A),bool,finite_finite2(A),A6))
            & pp(aa(set(B),bool,finite_finite2(B),B6)) )
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A6,B6)) = zero_zero(nat) ) ) ) ).

% card_Plus_conv_if
tff(fact_6805_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa2: list(A),Y: A] :
          ( ( arg_min_list(A,B,X,Xa2) = Y )
         => ( ! [X3: A] :
                ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
               => ( Y != X3 ) )
           => ( ! [X3: A,Y3: A,Zs2: list(A)] :
                  ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)) )
                 => ( Y != if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X3)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)))),X3,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2))) ) )
             => ~ ( ( Xa2 = nil(A) )
                 => ( Y != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_6806_enat__def,axiom,
    ! [N: nat] : extended_enat2(N) = extended_Abs_enat(aa(nat,option(nat),some(nat),N)) ).

% enat_def
tff(fact_6807_option_Othe__def,axiom,
    ! [A: $tType,Option: option(A)] : aa(option(A),A,the2(A),Option) = aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),undefined(A)),aTP_Lamp_abo(A,A)),Option) ).

% option.the_def
tff(fact_6808_nth__item_Opsimps_I1_J,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ( accp(nat,nth_item_rel,zero_zero(nat))
       => ( nth_item(A,zero_zero(nat)) = undefined(set(old_node(A,product_unit))) ) ) ) ).

% nth_item.psimps(1)
tff(fact_6809_arg__min__list_Opelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa2: list(A),Y: A] :
          ( ( arg_min_list(A,B,X,Xa2) = Y )
         => ( accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),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)),X),Xa2))
           => ( ! [X3: A] :
                  ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
                 => ( ( Y = X3 )
                   => ~ accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),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)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))) ) )
             => ( ! [X3: A,Y3: A,Zs2: list(A)] :
                    ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)) )
                   => ( ( Y = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X3)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)))),X3,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2))) )
                     => ~ accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),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)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)))) ) )
               => ~ ( ( Xa2 = nil(A) )
                   => ( ( Y = undefined(A) )
                     => ~ accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),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)),X),nil(A))) ) ) ) ) ) ) ) ).

% arg_min_list.pelims
tff(fact_6810_nth__item_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ( nth_item(A,zero_zero(nat)) = undefined(set(old_node(A,product_unit))) ) ) ).

% nth_item.simps(1)
tff(fact_6811_nth__item_Opelims,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [X: nat,Y: set(old_node(A,product_unit))] :
          ( ( nth_item(A,X) = Y )
         => ( accp(nat,nth_item_rel,X)
           => ( ( ( X = zero_zero(nat) )
               => ( ( Y = undefined(set(old_node(A,product_unit))) )
                 => ~ accp(nat,nth_item_rel,zero_zero(nat)) ) )
             => ~ ! [N3: nat] :
                    ( ( X = aa(nat,nat,suc,N3) )
                   => ( ( Y = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_agz(nat,set(old_node(A,product_unit))),aTP_Lamp_ahd(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N3)) )
                     => ~ accp(nat,nth_item_rel,aa(nat,nat,suc,N3)) ) ) ) ) ) ) ).

% nth_item.pelims
tff(fact_6812_nth__item_Opsimps_I2_J,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [N: nat] :
          ( accp(nat,nth_item_rel,aa(nat,nat,suc,N))
         => ( nth_item(A,aa(nat,nat,suc,N)) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_agz(nat,set(old_node(A,product_unit))),aTP_Lamp_ahd(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N)) ) ) ) ).

% nth_item.psimps(2)
tff(fact_6813_nth__item_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [N: nat] : nth_item(A,aa(nat,nat,suc,N)) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_agz(nat,set(old_node(A,product_unit))),aTP_Lamp_ahd(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N)) ) ).

% nth_item.simps(2)
tff(fact_6814_nth__item_Oelims,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [X: nat,Y: set(old_node(A,product_unit))] :
          ( ( nth_item(A,X) = Y )
         => ( ( ( X = zero_zero(nat) )
             => ( Y != undefined(set(old_node(A,product_unit))) ) )
           => ~ ! [N3: nat] :
                  ( ( X = aa(nat,nat,suc,N3) )
                 => ( Y != aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_agz(nat,set(old_node(A,product_unit))),aTP_Lamp_ahd(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N3)) ) ) ) ) ) ).

% nth_item.elims
tff(fact_6815_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F3: fun(nat,real),M5: nat] :
      ( bfun(nat,real,aa(nat,fun(nat,real),aTP_Lamp_ahe(fun(nat,real),fun(nat,fun(nat,real)),F3),M5),at_top(nat))
     => ( ! [M: nat,N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),M))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N3))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,N3)),aa(nat,real,F3,M))) ) )
       => topolo6863149650580417670ergent(real,F3) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_6816_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F3: fun(nat,real),M5: nat] :
      ( bfun(nat,real,aa(nat,fun(nat,real),aTP_Lamp_ahe(fun(nat,real),fun(nat,fun(nat,real)),F3),M5),at_top(nat))
     => ( ! [M: nat,N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),M))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N3))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,M)),aa(nat,real,F3,N3))) ) )
       => topolo6863149650580417670ergent(real,F3) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_6817_lim__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(nat,A),X: A] :
          ( topolo6863149650580417670ergent(A,F3)
         => ( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N3)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),topolo3827282254853284352ce_Lim(nat,A,at_top(nat),F3)),X)) ) ) ) ).

% lim_le
tff(fact_6818_convergent__Suc__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,aTP_Lamp_tm(fun(nat,A),fun(nat,A),F3))
        <=> topolo6863149650580417670ergent(A,F3) ) ) ).

% convergent_Suc_iff
tff(fact_6819_convergent__mult,axiom,
    ! [A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [X6: fun(nat,A),Y6: fun(nat,A)] :
          ( topolo6863149650580417670ergent(A,X6)
         => ( topolo6863149650580417670ergent(A,Y6)
           => topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ahf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),X6),Y6)) ) ) ) ).

% convergent_mult
tff(fact_6820_convergent__mult__const__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F3: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_tk(A,fun(fun(nat,A),fun(nat,A)),C3),F3))
          <=> topolo6863149650580417670ergent(A,F3) ) ) ) ).

% convergent_mult_const_iff
tff(fact_6821_convergent__mult__const__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F3: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_tj(A,fun(fun(nat,A),fun(nat,A)),C3),F3))
          <=> topolo6863149650580417670ergent(A,F3) ) ) ) ).

% convergent_mult_const_right_iff
tff(fact_6822_Bseq__mono__convergent,axiom,
    ! [X6: fun(nat,real)] :
      ( bfun(nat,real,X6,at_top(nat))
     => ( ! [M: nat,N3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N3))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,X6,M)),aa(nat,real,X6,N3))) )
       => topolo6863149650580417670ergent(real,X6) ) ) ).

% Bseq_mono_convergent
tff(fact_6823_iterates_Osimps,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: A,F3: fun(A,A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),comple6359979572994053840erates(A,F3)))
        <=> ( ? [X4: A] :
                ( ( A3 = aa(A,A,F3,X4) )
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),comple6359979572994053840erates(A,F3))) )
            | ? [M9: set(A)] :
                ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M9) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M9)
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),M9))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),comple6359979572994053840erates(A,F3))) ) ) ) ) ) ).

% iterates.simps
tff(fact_6824_iterates_OSup,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [M5: set(A),F3: fun(A,A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),M5)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),M5))
               => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),comple6359979572994053840erates(A,F3))) )
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,complete_Sup_Sup(A),M5)),comple6359979572994053840erates(A,F3))) ) ) ) ).

% iterates.Sup
tff(fact_6825_iterates_Ocases,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: A,F3: fun(A,A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),comple6359979572994053840erates(A,F3)))
         => ( ! [X3: A] :
                ( ( A3 = aa(A,A,F3,X3) )
               => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),comple6359979572994053840erates(A,F3))) )
           => ~ ! [M8: set(A)] :
                  ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M8) )
                 => ( comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                   => ~ ! [X5: A] :
                          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),M8))
                         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),comple6359979572994053840erates(A,F3))) ) ) ) ) ) ) ).

% iterates.cases
tff(fact_6826_chain__iterates,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A)] :
          ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F3)
         => comple1602240252501008431_chain(A,ord_less_eq(A),comple6359979572994053840erates(A,F3)) ) ) ).

% chain_iterates
tff(fact_6827_sorted__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),map(B,A,F3,linorder_sort_key(B,A,F3,Xs))) ) ).

% sorted_sort_key
tff(fact_6828_length__sort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(list(B),nat,size_size(list(B)),linorder_sort_key(B,A,F3,Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ) ).

% length_sort
tff(fact_6829_iterates__le__f,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: A,F3: fun(A,A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),comple6359979572994053840erates(A,F3)))
         => ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F3)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,F3,X))) ) ) ) ).

% iterates_le_f
tff(fact_6830_sorted__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),linorder_sort_key(A,A,aTP_Lamp_ni(A,A),Xs)) ) ).

% sorted_sort
tff(fact_6831_sorted__sort__id,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( linorder_sort_key(A,A,aTP_Lamp_ni(A,A),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_6832_fixp__induct,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P2: fun(A,bool),F3: fun(A,A)] :
          ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P2)
         => ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F3)
           => ( pp(aa(A,bool,P2,aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A)))))
             => ( ! [X3: A] :
                    ( pp(aa(A,bool,P2,X3))
                   => pp(aa(A,bool,P2,aa(A,A,F3,X3))) )
               => pp(aa(A,bool,P2,comple115746919287870866o_fixp(A,F3))) ) ) ) ) ) ).

% fixp_induct
tff(fact_6833_iterates__fixp,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A)] :
          ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F3)
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),comple115746919287870866o_fixp(A,F3)),comple6359979572994053840erates(A,F3))) ) ) ).

% iterates_fixp
tff(fact_6834_fixp__lowerbound,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A),Z2: A] :
          ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,Z2)),Z2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),comple115746919287870866o_fixp(A,F3)),Z2)) ) ) ) ).

% fixp_lowerbound
tff(fact_6835_fixp__unfold,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A)] :
          ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F3)
         => ( comple115746919287870866o_fixp(A,F3) = aa(A,A,F3,comple115746919287870866o_fixp(A,F3)) ) ) ) ).

% fixp_unfold
tff(fact_6836_length__splice,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),splice(A,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_splice
tff(fact_6837_map__conv__bind__option,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X: option(B)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F3),X) = aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),X),aa(fun(B,A),fun(B,option(A)),comp(A,option(A),B,some(A)),F3)) ).

% map_conv_bind_option
tff(fact_6838_bind__assoc,axiom,
    ! [C: $tType,A: $tType,B: $tType,X: option(C),F3: fun(C,option(B)),G3: fun(B,option(A))] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),aa(fun(C,option(B)),option(B),aa(option(C),fun(fun(C,option(B)),option(B)),bind(C,B),X),F3)),G3) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind(C,A),X),aa(fun(B,option(A)),fun(C,option(A)),aTP_Lamp_ahg(fun(C,option(B)),fun(fun(B,option(A)),fun(C,option(A))),F3),G3)) ).

% bind_assoc
tff(fact_6839_bind__runit,axiom,
    ! [A: $tType,X: option(A)] : aa(fun(A,option(A)),option(A),aa(option(A),fun(fun(A,option(A)),option(A)),bind(A,A),X),some(A)) = X ).

% bind_runit
tff(fact_6840_bind__rzero,axiom,
    ! [B: $tType,A: $tType,X: option(B)] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),X),aTP_Lamp_ahh(B,option(A))) = none(A) ).

% bind_rzero
tff(fact_6841_bind__eq__None__conv,axiom,
    ! [B: $tType,A: $tType,A3: option(B),F3: fun(B,option(A))] :
      ( ( aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),A3),F3) = none(A) )
    <=> ( ( A3 = none(B) )
        | ( aa(B,option(A),F3,aa(option(B),B,the2(B),A3)) = none(A) ) ) ) ).

% bind_eq_None_conv
tff(fact_6842_bind_Obind__lzero,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B))] : aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),none(A)),F3) = none(B) ).

% bind.bind_lzero
tff(fact_6843_bind_Obind__lunit,axiom,
    ! [B: $tType,A: $tType,X: A,F3: fun(A,option(B))] : aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),aa(A,option(A),some(A),X)),F3) = aa(A,option(B),F3,X) ).

% bind.bind_lunit
tff(fact_6844_Option_Obind__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y: option(A),F3: fun(A,option(B)),G3: fun(A,option(B))] :
      ( ( X = Y )
     => ( ! [A5: A] :
            ( ( Y = aa(A,option(A),some(A),A5) )
           => ( aa(A,option(B),F3,A5) = aa(A,option(B),G3,A5) ) )
       => ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),X),F3) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Y),G3) ) ) ) ).

% Option.bind_cong
tff(fact_6845_bind__eq__Some__conv,axiom,
    ! [B: $tType,A: $tType,F3: option(B),G3: fun(B,option(A)),X: A] :
      ( ( aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),F3),G3) = aa(A,option(A),some(A),X) )
    <=> ? [Y5: B] :
          ( ( F3 = aa(B,option(B),some(B),Y5) )
          & ( aa(B,option(A),G3,Y5) = aa(A,option(A),some(A),X) ) ) ) ).

% bind_eq_Some_conv
tff(fact_6846_bind__option__cong__code,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y: option(A),F3: fun(A,option(B))] :
      ( ( X = Y )
     => ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),X),F3) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Y),F3) ) ) ).

% bind_option_cong_code
tff(fact_6847_bind__split,axiom,
    ! [A: $tType,B: $tType,P2: fun(option(A),bool),M2: option(B),F3: fun(B,option(A))] :
      ( pp(aa(option(A),bool,P2,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),M2),F3)))
    <=> ( ( ( M2 = none(B) )
         => pp(aa(option(A),bool,P2,none(A))) )
        & ! [V4: B] :
            ( ( M2 = aa(B,option(B),some(B),V4) )
           => pp(aa(option(A),bool,P2,aa(B,option(A),F3,V4))) ) ) ) ).

% bind_split
tff(fact_6848_bind__split__asm,axiom,
    ! [A: $tType,B: $tType,P2: fun(option(A),bool),M2: option(B),F3: fun(B,option(A))] :
      ( pp(aa(option(A),bool,P2,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),M2),F3)))
    <=> ~ ( ( ( M2 = none(B) )
            & ~ pp(aa(option(A),bool,P2,none(A))) )
          | ? [X4: B] :
              ( ( M2 = aa(B,option(B),some(B),X4) )
              & ~ pp(aa(option(A),bool,P2,aa(B,option(A),F3,X4))) ) ) ) ).

% bind_split_asm
tff(fact_6849_bind__map__option,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,B),X: option(C),G3: fun(B,option(A))] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),aa(option(C),option(B),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F3),X)),G3) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind(C,A),X),aa(fun(C,B),fun(C,option(A)),comp(B,option(A),C,G3),F3)) ).

% bind_map_option
tff(fact_6850_map__option__bind,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(B,A),X: option(C),G3: fun(C,option(B))] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F3),aa(fun(C,option(B)),option(B),aa(option(C),fun(fun(C,option(B)),option(B)),bind(C,B),X),G3)) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind(C,A),X),aa(fun(C,option(B)),fun(C,option(A)),comp(option(B),option(A),C,aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F3)),G3)) ).

% map_option_bind
tff(fact_6851_set__bind__option,axiom,
    ! [A: $tType,B: $tType,X: option(B),F3: fun(B,option(A))] : aa(option(A),set(A),set_option(A),aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),X),F3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(B,set(A),aa(fun(B,option(A)),fun(B,set(A)),comp(option(A),set(A),B,set_option(A)),F3),aa(option(B),set(B),set_option(B),X))) ).

% set_bind_option
tff(fact_6852_continuous__attains__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S2: set(A),F3: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S2,F3)
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                  & ! [Xa: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),S2))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Xa)),aa(A,B,F3,X3))) ) ) ) ) ) ) ).

% continuous_attains_sup
tff(fact_6853_elem__set,axiom,
    ! [A: $tType,X: A,Xo: option(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(option(A),set(A),set_option(A),Xo)))
    <=> ( Xo = aa(A,option(A),some(A),X) ) ) ).

% elem_set
tff(fact_6854_set__empty__eq,axiom,
    ! [A: $tType,Xo: option(A)] :
      ( ( aa(option(A),set(A),set_option(A),Xo) = bot_bot(set(A)) )
    <=> ( Xo = none(A) ) ) ).

% set_empty_eq
tff(fact_6855_bind__option__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y: option(A),F3: fun(A,option(B)),G3: fun(A,option(B))] :
      ( ( X = Y )
     => ( ! [Z: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(option(A),set(A),set_option(A),Y)))
           => ( aa(A,option(B),F3,Z) = aa(A,option(B),G3,Z) ) )
       => ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),X),F3) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Y),G3) ) ) ) ).

% bind_option_cong
tff(fact_6856_option_Osimps_I14_J,axiom,
    ! [A: $tType] : aa(option(A),set(A),set_option(A),none(A)) = bot_bot(set(A)) ).

% option.simps(14)
tff(fact_6857_ospec,axiom,
    ! [A: $tType,A6: option(A),P2: fun(A,bool),X: A] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(option(A),set(A),set_option(A),A6)))
         => pp(aa(A,bool,P2,X3)) )
     => ( ( A6 = aa(A,option(A),some(A),X) )
       => pp(aa(A,bool,P2,X)) ) ) ).

% ospec
tff(fact_6858_option_Oset__intros,axiom,
    ! [A: $tType,X2: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(option(A),set(A),set_option(A),aa(A,option(A),some(A),X2)))) ).

% option.set_intros
tff(fact_6859_option_Oset__cases,axiom,
    ! [A: $tType,E3: A,A3: option(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),E3),aa(option(A),set(A),set_option(A),A3)))
     => ( A3 = aa(A,option(A),some(A),E3) ) ) ).

% option.set_cases
tff(fact_6860_option_Omap__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Ya: option(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( X = Ya )
     => ( ! [Z: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(option(A),set(A),set_option(A),Ya)))
           => ( aa(A,B,F3,Z) = aa(A,B,G3,Z) ) )
       => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G3),Ya) ) ) ) ).

% option.map_cong
tff(fact_6861_option_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,X: option(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ! [Z: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(option(A),set(A),set_option(A),X)))
         => ( aa(A,B,F3,Z) = aa(A,B,G3,Z) ) )
     => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G3),X) ) ) ).

% option.map_cong0
tff(fact_6862_option_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Xa2: option(A),F3: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z: A,Za2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(option(A),set(A),set_option(A),X)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Za2),aa(option(A),set(A),set_option(A),Xa2)))
           => ( ( aa(A,B,F3,Z) = aa(A,B,Fa,Za2) )
             => ( Z = Za2 ) ) ) )
     => ( ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),Fa),Xa2) )
       => ( X = Xa2 ) ) ) ).

% option.inj_map_strong
tff(fact_6863_map__option__idI,axiom,
    ! [A: $tType,X: option(A),F3: fun(A,A)] :
      ( ! [Y3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),aa(option(A),set(A),set_option(A),X)))
         => ( aa(A,A,F3,Y3) = Y3 ) )
     => ( aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),F3),X) = X ) ) ).

% map_option_idI
tff(fact_6864_option_Oset__sel,axiom,
    ! [A: $tType,A3: option(A)] :
      ( ( A3 != none(A) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(option(A),A,the2(A),A3)),aa(option(A),set(A),set_option(A),A3))) ) ).

% option.set_sel
tff(fact_6865_option_Oset__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),V3: option(A)] : aa(option(B),set(B),set_option(B),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),V3)) = image2(A,B,F3,aa(option(A),set(A),set_option(A),V3)) ).

% option.set_map
tff(fact_6866_compact__attains__sup,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A)] :
          ( topolo2193935891317330818ompact(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),S3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa),X3)) ) ) ) ) ) ).

% compact_attains_sup
tff(fact_6867_compact__attains__inf,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A)] :
          ( topolo2193935891317330818ompact(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
                & ! [Xa: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),S3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa)) ) ) ) ) ) ).

% compact_attains_inf
tff(fact_6868_option_Osimps_I15_J,axiom,
    ! [A: $tType,X2: A] : aa(option(A),set(A),set_option(A),aa(A,option(A),some(A),X2)) = aa(set(A),set(A),insert(A,X2),bot_bot(set(A))) ).

% option.simps(15)
tff(fact_6869_continuous__attains__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S2: set(A),F3: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S2,F3)
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                  & ! [Xa: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),S2))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Xa))) ) ) ) ) ) ) ).

% continuous_attains_inf
tff(fact_6870_option_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),A3),B2))
    <=> ? [Z3: option(product_prod(A,B))] :
          ( pp(aa(set(option(product_prod(A,B))),bool,aa(option(product_prod(A,B)),fun(set(option(product_prod(A,B))),bool),member(option(product_prod(A,B))),Z3),aa(fun(option(product_prod(A,B)),bool),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_ahi(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),R2))))
          & ( aa(option(product_prod(A,B)),option(A),aa(fun(product_prod(A,B),A),fun(option(product_prod(A,B)),option(A)),map_option(product_prod(A,B),A),product_fst(A,B)),Z3) = A3 )
          & ( aa(option(product_prod(A,B)),option(B),aa(fun(product_prod(A,B),B),fun(option(product_prod(A,B)),option(B)),map_option(product_prod(A,B),B),product_snd(A,B)),Z3) = B2 ) ) ) ).

% option.in_rel
tff(fact_6871_combine__options__def,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),X: option(A),Y: option(A)] : combine_options(A,F3,X,Y) = aa(option(A),option(A),aa(fun(A,option(A)),fun(option(A),option(A)),aa(option(A),fun(fun(A,option(A)),fun(option(A),option(A))),case_option(option(A),A),Y),aa(option(A),fun(A,option(A)),aTP_Lamp_ahk(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),F3),Y)),X) ).

% combine_options_def
tff(fact_6872_rel__option__None1,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),X: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),P2),none(A)),X))
    <=> ( X = none(B) ) ) ).

% rel_option_None1
tff(fact_6873_rel__option__None2,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,bool)),X: option(A)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),P2),X),none(B)))
    <=> ( X = none(A) ) ) ).

% rel_option_None2
tff(fact_6874_combine__options__simps_I3_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),A3: A,B2: A] : combine_options(A,F3,aa(A,option(A),some(A),A3),aa(A,option(A),some(A),B2)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),F3,A3),B2)) ).

% combine_options_simps(3)
tff(fact_6875_combine__options__simps_I1_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Y: option(A)] : combine_options(A,F3,none(A),Y) = Y ).

% combine_options_simps(1)
tff(fact_6876_combine__options__simps_I2_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),X: option(A)] : combine_options(A,F3,X,none(A)) = X ).

% combine_options_simps(2)
tff(fact_6877_rel__option__reflI,axiom,
    ! [A: $tType,Y: option(A),P2: fun(A,fun(A,bool))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(option(A),set(A),set_option(A),Y)))
         => pp(aa(A,bool,aa(A,fun(A,bool),P2,X3),X3)) )
     => pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),P2),Y),Y)) ) ).

% rel_option_reflI
tff(fact_6878_option_Orel__refl__strong,axiom,
    ! [A: $tType,X: option(A),Ra2: fun(A,fun(A,bool))] :
      ( ! [Z: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(option(A),set(A),set_option(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),Ra2,Z),Z)) )
     => pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),Ra2),X),X)) ) ).

% option.rel_refl_strong
tff(fact_6879_option_Orel__mono__strong,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X: option(A),Y: option(B),Ra2: fun(A,fun(B,bool))] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),X),Y))
     => ( ! [Z: A,Yb: B] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(option(A),set(A),set_option(A),X)))
           => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Yb),aa(option(B),set(B),set_option(B),Y)))
             => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,Z),Yb))
               => pp(aa(B,bool,aa(A,fun(B,bool),Ra2,Z),Yb)) ) ) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),Ra2),X),Y)) ) ) ).

% option.rel_mono_strong
tff(fact_6880_option_Orel__cong,axiom,
    ! [A: $tType,B: $tType,X: option(A),Ya: option(A),Y: option(B),Xa2: option(B),R2: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
      ( ( X = Ya )
     => ( ( Y = Xa2 )
       => ( ! [Z: A,Yb: B] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(option(A),set(A),set_option(A),Ya)))
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Yb),aa(option(B),set(B),set_option(B),Xa2)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),R2,Z),Yb))
                <=> pp(aa(B,bool,aa(A,fun(B,bool),Ra2,Z),Yb)) ) ) )
         => ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),X),Y))
          <=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),Ra2),Ya),Xa2)) ) ) ) ) ).

% option.rel_cong
tff(fact_6881_option_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R2),Ra2))
     => pp(aa(fun(option(A),fun(option(B),bool)),bool,aa(fun(option(A),fun(option(B),bool)),fun(fun(option(A),fun(option(B),bool)),bool),ord_less_eq(fun(option(A),fun(option(B),bool))),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),Ra2))) ) ).

% option.rel_mono
tff(fact_6882_option_Orec__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,S3: fun(C,fun(D,bool)),R2: fun(A,fun(B,bool))] : pp(aa(fun(D,fun(fun(B,D),fun(option(B),D))),bool,aa(fun(C,fun(fun(A,C),fun(option(A),C))),fun(fun(D,fun(fun(B,D),fun(option(B),D))),bool),bNF_rel_fun(C,D,fun(fun(A,C),fun(option(A),C)),fun(fun(B,D),fun(option(B),D)),S3,bNF_rel_fun(fun(A,C),fun(B,D),fun(option(A),C),fun(option(B),D),bNF_rel_fun(A,B,C,D,R2,S3),bNF_rel_fun(option(A),option(B),C,D,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),S3))),rec_option(C,A)),rec_option(D,B))) ).

% option.rec_transfer
tff(fact_6883_option_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I2: fun(A,C),X: option(A),Y: option(B)] :
      ( pp(aa(option(B),bool,aa(option(C),fun(option(B),bool),aa(fun(C,fun(B,bool)),fun(option(C),fun(option(B),bool)),rel_option(C,B),Sb),aa(option(A),option(C),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),I2),X)),Y))
    <=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_ahl(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I2)),X),Y)) ) ).

% option.rel_map(1)
tff(fact_6884_option_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: option(A),G3: fun(B,C),Y: option(B)] :
      ( pp(aa(option(C),bool,aa(option(A),fun(option(C),bool),aa(fun(A,fun(C,bool)),fun(option(A),fun(option(C),bool)),rel_option(A,C),Sa),X),aa(option(B),option(C),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),G3),Y)))
    <=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_ahm(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G3)),X),Y)) ) ).

% option.rel_map(2)
tff(fact_6885_option_Omap__transfer,axiom,
    ! [A: $tType,B: $tType,F: $tType,E: $tType,Rb2: fun(A,fun(E,bool)),Sd: fun(B,fun(F,bool))] : pp(aa(fun(fun(E,F),fun(option(E),option(F))),bool,aa(fun(fun(A,B),fun(option(A),option(B))),fun(fun(fun(E,F),fun(option(E),option(F))),bool),bNF_rel_fun(fun(A,B),fun(E,F),fun(option(A),option(B)),fun(option(E),option(F)),bNF_rel_fun(A,E,B,F,Rb2,Sd),bNF_rel_fun(option(A),option(E),option(B),option(F),aa(fun(A,fun(E,bool)),fun(option(A),fun(option(E),bool)),rel_option(A,E),Rb2),aa(fun(B,fun(F,bool)),fun(option(B),fun(option(F),bool)),rel_option(B,F),Sd))),map_option(A,B)),map_option(E,F))) ).

% option.map_transfer
tff(fact_6886_option_Odisc__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(option(B),bool),bool,aa(fun(option(A),bool),fun(fun(option(B),bool),bool),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),fequal(bool)),aTP_Lamp_ahn(option(A),bool)),aTP_Lamp_aho(option(B),bool))) ).

% option.disc_transfer(1)
tff(fact_6887_option_Odisc__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(option(B),bool),bool,aa(fun(option(A),bool),fun(fun(option(B),bool),bool),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),fequal(bool)),aTP_Lamp_ahp(option(A),bool)),aTP_Lamp_ahq(option(B),bool))) ).

% option.disc_transfer(2)
tff(fact_6888_option_Octr__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(B,option(B)),bool,aa(fun(A,option(A)),fun(fun(B,option(B)),bool),bNF_rel_fun(A,B,option(A),option(B),R2,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2)),some(A)),some(B))) ).

% option.ctr_transfer(2)
tff(fact_6889_option_Orel__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: fun(A,fun(C,bool)),Sc: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool))),bool,aa(fun(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool))),fun(fun(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool))),bool),bNF_rel_fun(fun(A,fun(B,bool)),fun(C,fun(D,bool)),fun(option(A),fun(option(B),bool)),fun(option(C),fun(option(D),bool)),bNF_rel_fun(A,C,fun(B,bool),fun(D,bool),Sa,bNF_rel_fun(B,D,bool,bool,Sc,fequal(bool))),bNF_rel_fun(option(A),option(C),fun(option(B),bool),fun(option(D),bool),aa(fun(A,fun(C,bool)),fun(option(A),fun(option(C),bool)),rel_option(A,C),Sa),bNF_rel_fun(option(B),option(D),bool,bool,aa(fun(B,fun(D,bool)),fun(option(B),fun(option(D),bool)),rel_option(B,D),Sc),fequal(bool)))),rel_option(A,B)),rel_option(C,D))) ).

% option.rel_transfer
tff(fact_6890_option_Obi__total__rel,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] :
      ( bi_total(A,B,R2)
     => bi_total(option(A),option(B),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2)) ) ).

% option.bi_total_rel
tff(fact_6891_option_Orel__inject_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X2: A,Y2: B] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),aa(A,option(A),some(A),X2)),aa(B,option(B),some(B),Y2)))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),R2,X2),Y2)) ) ).

% option.rel_inject(2)
tff(fact_6892_option_Orel__intros_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X2: A,Y2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R2,X2),Y2))
     => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),aa(A,option(A),some(A),X2)),aa(B,option(B),some(B),Y2))) ) ).

% option.rel_intros(2)
tff(fact_6893_option__rel__Some1,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool)),X: A,Y: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A6),aa(A,option(A),some(A),X)),Y))
    <=> ? [Y8: B] :
          ( ( Y = aa(B,option(B),some(B),Y8) )
          & pp(aa(B,bool,aa(A,fun(B,bool),A6,X),Y8)) ) ) ).

% option_rel_Some1
tff(fact_6894_option__rel__Some2,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool)),X: option(A),Y: B] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A6),X),aa(B,option(B),some(B),Y)))
    <=> ? [X9: A] :
          ( ( X = aa(A,option(A),some(A),X9) )
          & pp(aa(B,bool,aa(A,fun(B,bool),A6,X9),Y)) ) ) ).

% option_rel_Some2
tff(fact_6895_option_Octr__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),none(A)),none(B))) ).

% option.ctr_transfer(1)
tff(fact_6896_option_Orel__induct,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X: option(A),Y: option(B),Q: fun(option(A),fun(option(B),bool))] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),X),Y))
     => ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),Q,none(A)),none(B)))
       => ( ! [A22: A,B23: B] :
              ( pp(aa(B,bool,aa(A,fun(B,bool),R2,A22),B23))
             => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),Q,aa(A,option(A),some(A),A22)),aa(B,option(B),some(B),B23))) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),Q,X),Y)) ) ) ) ).

% option.rel_induct
tff(fact_6897_option_Orel__cases,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),A3),B2))
     => ( ( ( A3 = none(A) )
         => ( B2 != none(B) ) )
       => ~ ! [X3: A] :
              ( ( A3 = aa(A,option(A),some(A),X3) )
             => ! [Y3: B] :
                  ( ( B2 = aa(B,option(B),some(B),Y3) )
                 => ~ pp(aa(B,bool,aa(A,fun(B,bool),R2,X3),Y3)) ) ) ) ) ).

% option.rel_cases
tff(fact_6898_option_Orel__distinct_I1_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),Y2: B] : ~ pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),none(A)),aa(B,option(B),some(B),Y2))) ).

% option.rel_distinct(1)
tff(fact_6899_option_Orel__distinct_I2_J,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),Y2: A] : ~ pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),aa(A,option(A),some(A),Y2)),none(B))) ).

% option.rel_distinct(2)
tff(fact_6900_option_Orel__sel,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),A3),B2))
    <=> ( ( ( A3 = none(A) )
        <=> ( B2 = none(B) ) )
        & ( ( A3 != none(A) )
         => ( ( B2 != none(B) )
           => pp(aa(B,bool,aa(A,fun(B,bool),R2,aa(option(A),A,the2(A),A3)),aa(option(B),B,the2(B),B2))) ) ) ) ) ).

% option.rel_sel
tff(fact_6901_option_Orel__eq,axiom,
    ! [A: $tType] : aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),fequal(A)) = fequal(option(A)) ).

% option.rel_eq
tff(fact_6902_option_Orel__refl,axiom,
    ! [B: $tType,Ra2: fun(B,fun(B,bool)),X: option(B)] :
      ( ! [X3: B] : pp(aa(B,bool,aa(B,fun(B,bool),Ra2,X3),X3))
     => pp(aa(option(B),bool,aa(option(B),fun(option(B),bool),aa(fun(B,fun(B,bool)),fun(option(B),fun(option(B),bool)),rel_option(B,B),Ra2),X),X)) ) ).

% option.rel_refl
tff(fact_6903_combine__options__assoc,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),X: option(A),Y: option(A),Z2: option(A)] :
      ( ! [X3: A,Y3: A,Z: A] : aa(A,A,aa(A,fun(A,A),F3,aa(A,A,aa(A,fun(A,A),F3,X3),Y3)),Z) = aa(A,A,aa(A,fun(A,A),F3,X3),aa(A,A,aa(A,fun(A,A),F3,Y3),Z))
     => ( combine_options(A,F3,combine_options(A,F3,X,Y),Z2) = combine_options(A,F3,X,combine_options(A,F3,Y,Z2)) ) ) ).

% combine_options_assoc
tff(fact_6904_combine__options__commute,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),X: option(A),Y: option(A)] :
      ( ! [X3: A,Y3: A] : aa(A,A,aa(A,fun(A,A),F3,X3),Y3) = aa(A,A,aa(A,fun(A,A),F3,Y3),X3)
     => ( combine_options(A,F3,X,Y) = combine_options(A,F3,Y,X) ) ) ).

% combine_options_commute
tff(fact_6905_combine__options__left__commute,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Y: option(A),X: option(A),Z2: option(A)] :
      ( ! [X3: A,Y3: A] : aa(A,A,aa(A,fun(A,A),F3,X3),Y3) = aa(A,A,aa(A,fun(A,A),F3,Y3),X3)
     => ( ! [X3: A,Y3: A,Z: A] : aa(A,A,aa(A,fun(A,A),F3,aa(A,A,aa(A,fun(A,A),F3,X3),Y3)),Z) = aa(A,A,aa(A,fun(A,A),F3,X3),aa(A,A,aa(A,fun(A,A),F3,Y3),Z))
       => ( combine_options(A,F3,Y,combine_options(A,F3,X,Z2)) = combine_options(A,F3,X,combine_options(A,F3,Y,Z2)) ) ) ) ).

% combine_options_left_commute
tff(fact_6906_rel__option__inf,axiom,
    ! [B: $tType,A: $tType,A6: fun(A,fun(B,bool)),B6: fun(A,fun(B,bool))] : aa(fun(option(A),fun(option(B),bool)),fun(option(A),fun(option(B),bool)),aa(fun(option(A),fun(option(B),bool)),fun(fun(option(A),fun(option(B),bool)),fun(option(A),fun(option(B),bool))),inf_inf(fun(option(A),fun(option(B),bool))),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A6)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),B6)) = aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),A6),B6)) ).

% rel_option_inf
tff(fact_6907_option_Orel__transp,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] :
      ( transp(A,R2)
     => transp(option(A),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),R2)) ) ).

% option.rel_transp
tff(fact_6908_option_Orel__conversep,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : aa(fun(B,fun(A,bool)),fun(option(B),fun(option(A),bool)),rel_option(B,A),conversep(A,B,R2)) = conversep(option(A),option(B),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2)) ).

% option.rel_conversep
tff(fact_6909_option_Orel__flip,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),A3: option(B),B2: option(A)] :
      ( pp(aa(option(A),bool,aa(option(B),fun(option(A),bool),aa(fun(B,fun(A,bool)),fun(option(B),fun(option(A),bool)),rel_option(B,A),conversep(A,B,R2)),A3),B2))
    <=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),B2),A3)) ) ).

% option.rel_flip
tff(fact_6910_option_Ocase__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,S3: fun(C,fun(D,bool)),R2: fun(A,fun(B,bool))] : pp(aa(fun(D,fun(fun(B,D),fun(option(B),D))),bool,aa(fun(C,fun(fun(A,C),fun(option(A),C))),fun(fun(D,fun(fun(B,D),fun(option(B),D))),bool),bNF_rel_fun(C,D,fun(fun(A,C),fun(option(A),C)),fun(fun(B,D),fun(option(B),D)),S3,bNF_rel_fun(fun(A,C),fun(B,D),fun(option(A),C),fun(option(B),D),bNF_rel_fun(A,B,C,D,R2,S3),bNF_rel_fun(option(A),option(B),C,D,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),S3))),case_option(C,A)),case_option(D,B))) ).

% option.case_transfer
tff(fact_6911_option__bind__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A6: fun(A,fun(B,bool)),B6: fun(C,fun(D,bool))] : pp(aa(fun(option(B),fun(fun(B,option(D)),option(D))),bool,aa(fun(option(A),fun(fun(A,option(C)),option(C))),fun(fun(option(B),fun(fun(B,option(D)),option(D))),bool),bNF_rel_fun(option(A),option(B),fun(fun(A,option(C)),option(C)),fun(fun(B,option(D)),option(D)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A6),bNF_rel_fun(fun(A,option(C)),fun(B,option(D)),option(C),option(D),bNF_rel_fun(A,B,option(C),option(D),A6,aa(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool)),rel_option(C,D),B6)),aa(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool)),rel_option(C,D),B6))),bind(A,C)),bind(B,D))) ).

% option_bind_transfer
tff(fact_6912_rel__option__iff,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X: option(A),Y: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),X),Y))
    <=> pp(aa(product_prod(option(A),option(B)),bool,aa(fun(option(A),fun(option(B),bool)),fun(product_prod(option(A),option(B)),bool),product_case_prod(option(A),option(B),bool),aTP_Lamp_aht(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),R2)),aa(option(B),product_prod(option(A),option(B)),aa(option(A),fun(option(B),product_prod(option(A),option(B))),product_Pair(option(A),option(B)),X),Y))) ) ).

% rel_option_iff
tff(fact_6913_option_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] : aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2) = relcompp(option(A),option(product_prod(A,B)),option(B),conversep(option(product_prod(A,B)),option(A),bNF_Grp(option(product_prod(A,B)),option(A),aa(fun(option(product_prod(A,B)),bool),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_ahi(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),R2)),aa(fun(product_prod(A,B),A),fun(option(product_prod(A,B)),option(A)),map_option(product_prod(A,B),A),product_fst(A,B)))),bNF_Grp(option(product_prod(A,B)),option(B),aa(fun(option(product_prod(A,B)),bool),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_ahi(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),R2)),aa(fun(product_prod(A,B),B),fun(option(product_prod(A,B)),option(B)),map_option(product_prod(A,B),B),product_snd(A,B)))) ).

% option.rel_compp_Grp
tff(fact_6914_cut__def,axiom,
    ! [A: $tType,B: $tType,X: A,R2: set(product_prod(A,A)),F3: fun(A,B),X5: A] :
      ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),X)),R2))
       => ( aa(A,B,cut(A,B,F3,R2,X),X5) = aa(A,B,F3,X5) ) )
      & ( ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),X)),R2))
       => ( aa(A,B,cut(A,B,F3,R2,X),X5) = undefined(B) ) ) ) ).

% cut_def
tff(fact_6915_option_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,B)] : aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),bNF_Grp(A,B,A6,F3)) = bNF_Grp(option(A),option(B),aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_ahu(set(A),fun(option(A),bool),A6)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3)) ).

% option.rel_Grp
tff(fact_6916_option_Orel__compp,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: fun(A,fun(B,bool)),S3: fun(B,fun(C,bool))] : aa(fun(A,fun(C,bool)),fun(option(A),fun(option(C),bool)),rel_option(A,C),relcompp(A,B,C,R2,S3)) = relcompp(option(A),option(B),option(C),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),aa(fun(B,fun(C,bool)),fun(option(B),fun(option(C),bool)),rel_option(B,C),S3)) ).

% option.rel_compp
tff(fact_6917_relcompp__relcomp__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,B)),S2: set(product_prod(B,C)),X5: A,Xa: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),relcompp(A,B,C,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),R),aTP_Lamp_ahv(set(product_prod(B,C)),fun(B,fun(C,bool)),S2)),X5),Xa))
    <=> pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X5),Xa)),relcomp(A,B,C,R,S2))) ) ).

% relcompp_relcomp_eq
tff(fact_6918_cut__apply,axiom,
    ! [B: $tType,A: $tType,X: A,A3: A,R2: set(product_prod(A,A)),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3)),R2))
     => ( aa(A,B,cut(A,B,F3,R2,A3),X) = aa(A,B,F3,X) ) ) ).

% cut_apply
tff(fact_6919_cuts__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),R2: set(product_prod(A,A)),X: A,G3: fun(A,B)] :
      ( ( cut(A,B,F3,R2,X) = cut(A,B,G3,R2,X) )
    <=> ! [Y5: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X)),R2))
         => ( aa(A,B,F3,Y5) = aa(A,B,G3,Y5) ) ) ) ).

% cuts_eq
tff(fact_6920_relpowp_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,R2: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),R2) = relcompp(A,A,A,aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),R2),R2) ).

% relpowp.simps(2)
tff(fact_6921_relcomp__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,X5: set(product_prod(A,B)),Xa: set(product_prod(B,C))] : relcomp(A,B,C,X5,Xa) = aa(fun(product_prod(A,C),bool),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,bool)),fun(product_prod(A,C),bool),product_case_prod(A,C,bool),relcompp(A,B,C,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),X5),aTP_Lamp_ahv(set(product_prod(B,C)),fun(B,fun(C,bool)),Xa)))) ).

% relcomp_def
tff(fact_6922_pred__option__parametric,axiom,
    ! [A: $tType,B: $tType,A6: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(option(B),bool)),bool,aa(fun(fun(A,bool),fun(option(A),bool)),fun(fun(fun(B,bool),fun(option(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(option(A),bool),fun(option(B),bool),bNF_rel_fun(A,B,bool,bool,A6,fequal(bool)),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A6),fequal(bool))),pred_option(A)),pred_option(B))) ).

% pred_option_parametric
tff(fact_6923_option_Opred__transfer,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(option(B),bool)),bool,aa(fun(fun(A,bool),fun(option(A),bool)),fun(fun(fun(B,bool),fun(option(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(option(A),bool),fun(option(B),bool),bNF_rel_fun(A,B,bool,bool,R2,fequal(bool)),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),fequal(bool))),pred_option(A)),pred_option(B))) ).

% option.pred_transfer
tff(fact_6924_option_Opred__inject_I2_J,axiom,
    ! [A: $tType,P2: fun(A,bool),A3: A] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P2),aa(A,option(A),some(A),A3)))
    <=> pp(aa(A,bool,P2,A3)) ) ).

% option.pred_inject(2)
tff(fact_6925_option_Opred__mono,axiom,
    ! [A: $tType,P2: fun(A,bool),Pa: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P2),Pa))
     => pp(aa(fun(option(A),bool),bool,aa(fun(option(A),bool),fun(fun(option(A),bool),bool),ord_less_eq(fun(option(A),bool)),aa(fun(A,bool),fun(option(A),bool),pred_option(A),P2)),aa(fun(A,bool),fun(option(A),bool),pred_option(A),Pa))) ) ).

% option.pred_mono
tff(fact_6926_option_Opred__inject_I1_J,axiom,
    ! [A: $tType,P2: fun(A,bool)] : pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P2),none(A))) ).

% option.pred_inject(1)
tff(fact_6927_option_Opred__True,axiom,
    ! [A: $tType,X5: option(A)] : pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),aTP_Lamp_yf(A,bool)),X5)) ).

% option.pred_True
tff(fact_6928_option_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,X: option(A),Ya: option(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( X = Ya )
     => ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_ahw(fun(A,B),fun(fun(A,B),fun(A,bool)),F3),G3)),Ya))
       => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G3),Ya) ) ) ) ).

% option.map_cong_pred
tff(fact_6929_option_Opred__cong,axiom,
    ! [A: $tType,X: option(A),Ya: option(A),P2: fun(A,bool),Pa: fun(A,bool)] :
      ( ( X = Ya )
     => ( ! [Z: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(option(A),set(A),set_option(A),Ya)))
           => ( pp(aa(A,bool,P2,Z))
            <=> pp(aa(A,bool,Pa,Z)) ) )
       => ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P2),X))
        <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),Pa),Ya)) ) ) ) ).

% option.pred_cong
tff(fact_6930_option_Opred__mono__strong,axiom,
    ! [A: $tType,P2: fun(A,bool),X: option(A),Pa: fun(A,bool)] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P2),X))
     => ( ! [Z: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),aa(option(A),set(A),set_option(A),X)))
           => ( pp(aa(A,bool,P2,Z))
             => pp(aa(A,bool,Pa,Z)) ) )
       => pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),Pa),X)) ) ) ).

% option.pred_mono_strong
tff(fact_6931_option_Opred__set,axiom,
    ! [A: $tType,P2: fun(A,bool),X5: option(A)] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P2),X5))
    <=> ! [Xa3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),aa(option(A),set(A),set_option(A),X5)))
         => pp(aa(A,bool,P2,Xa3)) ) ) ).

% option.pred_set
tff(fact_6932_option_Opred__map,axiom,
    ! [B: $tType,A: $tType,Q: fun(B,bool),F3: fun(A,B),X: option(A)] :
      ( pp(aa(option(B),bool,aa(fun(B,bool),fun(option(B),bool),pred_option(B),Q),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),X)))
    <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),aa(fun(A,B),fun(A,bool),comp(B,bool,A,Q),F3)),X)) ) ).

% option.pred_map
tff(fact_6933_above__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_above(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).

% above_def
tff(fact_6934_old_Orec__unit__def,axiom,
    ! [T: $tType,X5: T,Xa: product_unit] : product_rec_unit(T,X5,Xa) = the(T,product_rec_set_unit(T,X5,Xa)) ).

% old.rec_unit_def
tff(fact_6935_length__removeAll__less,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,X,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_removeAll_less
tff(fact_6936_lcm__altdef__int,axiom,
    ! [A3: int,B2: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),A3),B2) = divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),A3)),aa(int,int,abs_abs(int),B2)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)) ).

% lcm_altdef_int
tff(fact_6937_lcm_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% lcm.bottom_right_bottom
tff(fact_6938_lcm_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% lcm.bottom_left_bottom
tff(fact_6939_lcm__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% lcm_eq_0_iff
tff(fact_6940_zero__eq__lcm__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_eq_lcm_iff
tff(fact_6941_lcm__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C3) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C3) ) ) ) ).

% lcm_mult_unit1
tff(fact_6942_lcm__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C3) ) ) ) ).

% lcm_mult_unit2
tff(fact_6943_length__removeAll__less__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,X,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_removeAll_less_eq
tff(fact_6944_prod__gcd__lcm__int,axiom,
    ! [M2: int,N: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),M2)),aa(int,int,abs_abs(int),N)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M2),N)),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M2),N)) ).

% prod_gcd_lcm_int
tff(fact_6945_Lcm__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite2(A),A6))
           => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A6) = finite_fold(A,A,gcd_lcm(A),one_one(A),A6) ) )
          & ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
           => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A6) = zero_zero(A) ) ) ) ) ).

% Lcm_fin.eq_fold
tff(fact_6946_Lcm__fin__def,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Lcm_fin(A) = bounde2362111253966948842tice_F(A,gcd_lcm(A),one_one(A),zero_zero(A)) ) ) ).

% Lcm_fin_def
tff(fact_6947_lcm__0__iff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M2),N) = zero_zero(nat) )
    <=> ( ( M2 = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% lcm_0_iff_nat
tff(fact_6948_lcm__1__iff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M2),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M2 = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% lcm_1_iff_nat
tff(fact_6949_Lcm__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A6) = zero_zero(A) ) ) ) ).

% Lcm_fin.infinite
tff(fact_6950_prod__gcd__lcm__nat,axiom,
    ! [M2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M2),N) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M2),N)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M2),N)) ).

% prod_gcd_lcm_nat
tff(fact_6951_lcm__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M2),N))) ) ) ).

% lcm_pos_nat
tff(fact_6952_lcm__code__integer,axiom,
    ! [A3: code_integer,B2: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_lcm(code_integer),A3),B2) = divide_divide(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),A3)),aa(code_integer,code_integer,abs_abs(code_integer),B2)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),A3),B2)) ).

% lcm_code_integer
tff(fact_6953_Lcm__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A6) = zero_zero(A) )
          <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),A6)) ) ) ) ).

% Lcm_fin_0_iff
tff(fact_6954_lcm__nat__def,axiom,
    ! [X: nat,Y: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),X),Y) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),Y),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y)) ).

% lcm_nat_def
tff(fact_6955_Lcm__eq__Max__nat,axiom,
    ! [M5: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),M5))
     => ( ( M5 != bot_bot(set(nat)) )
       => ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M5))
         => ( ! [M: nat,N3: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),M),M5))
               => ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N3),M5))
                 => pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N3)),M5)) ) )
           => ( gcd_Lcm(nat,M5) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),M5) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_6956_gcd__lcm,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ) ) ) ).

% gcd_lcm
tff(fact_6957_normalize__idem,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,normal6383669964737779283malize(A),A3)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% normalize_idem
tff(fact_6958_normalize__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% normalize_eq_0_iff
tff(fact_6959_normalize__0,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( aa(A,A,normal6383669964737779283malize(A),zero_zero(A)) = zero_zero(A) ) ) ).

% normalize_0
tff(fact_6960_lcm_Onormalize__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(A,A,normal6383669964737779283malize(A),zero_zero(A)) = zero_zero(A) ) ) ).

% lcm.normalize_bottom
tff(fact_6961_normalize__mult__normalize__right,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,normal6383669964737779283malize(A),B2))) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% normalize_mult_normalize_right
tff(fact_6962_normalize__mult__normalize__left,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% normalize_mult_normalize_left
tff(fact_6963_normalize__1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).

% normalize_1
tff(fact_6964_normalize__dvd__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,normal6383669964737779283malize(A),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% normalize_dvd_iff
tff(fact_6965_dvd__normalize__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,normal6383669964737779283malize(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_normalize_iff
tff(fact_6966_coprime__normalize__right__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,aa(A,A,normal6383669964737779283malize(A),B2))
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_normalize_right_iff
tff(fact_6967_coprime__normalize__left__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,normal6383669964737779283malize(A),A3),B2)
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_normalize_left_iff
tff(fact_6968_Lcm__eq__0__I__nat,axiom,
    ! [A6: set(nat)] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A6))
     => ( gcd_Lcm(nat,A6) = zero_zero(nat) ) ) ).

% Lcm_eq_0_I_nat
tff(fact_6969_Lcm__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Lcm(A,top_top(set(A))) = zero_zero(A) ) ) ).

% Lcm_UNIV
tff(fact_6970_gcd_Otop__right__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),zero_zero(A)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% gcd.top_right_normalize
tff(fact_6971_gcd_Otop__left__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),zero_zero(A)),A3) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% gcd.top_left_normalize
tff(fact_6972_Lcm__0__iff__nat,axiom,
    ! [A6: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A6))
     => ( ( gcd_Lcm(nat,A6) = zero_zero(nat) )
      <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A6)) ) ) ).

% Lcm_0_iff_nat
tff(fact_6973_normalize__mult__unit__right,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ) ) ).

% normalize_mult_unit_right
tff(fact_6974_normalize__mult__unit__left,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ).

% normalize_mult_unit_left
tff(fact_6975_gcd__mult__lcm,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).

% gcd_mult_lcm
tff(fact_6976_lcm__mult__gcd,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).

% lcm_mult_gcd
tff(fact_6977_lcm__mult__distrib_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C3)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ).

% lcm_mult_distrib'
tff(fact_6978_lcm__mult__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),A3)),C3)) ) ).

% lcm_mult_right
tff(fact_6979_lcm__mult__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ).

% lcm_mult_left
tff(fact_6980_Lcm__coprime_H,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A6: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),A6) != zero_zero(nat) )
         => ( ! [A5: A,B4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A6))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B4),A6))
                 => ( ( A5 != B4 )
                   => algebr8660921524188924756oprime(A,A5,B4) ) ) )
           => ( gcd_Lcm(A,A6) = aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_ahx(A,A)),A6)) ) ) ) ) ).

% Lcm_coprime'
tff(fact_6981_Lcm__mult,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A6: set(A),C3: A] :
          ( ( A6 != bot_bot(set(A)) )
         => ( gcd_Lcm(A,image2(A,A,aa(A,fun(A,A),times_times(A),C3),A6)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),gcd_Lcm(A,A6))) ) ) ) ).

% Lcm_mult
tff(fact_6982_dvd__normalize__div,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( aa(A,A,normal6383669964737779283malize(A),divide_divide(A,A3,B2)) = divide_divide(A,aa(A,A,normal6383669964737779283malize(A),A3),aa(A,A,normal6383669964737779283malize(A),B2)) ) ) ) ).

% dvd_normalize_div
tff(fact_6983_associated__iff__dvd,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ) ).

% associated_iff_dvd
tff(fact_6984_associated__eqI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
             => ( ( aa(A,A,normal6383669964737779283malize(A),B2) = B2 )
               => ( A3 = B2 ) ) ) ) ) ) ).

% associated_eqI
tff(fact_6985_associatedD2,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% associatedD2
tff(fact_6986_associatedD1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% associatedD1
tff(fact_6987_associatedI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ) ).

% associatedI
tff(fact_6988_gcd__mult__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ).

% gcd_mult_left
tff(fact_6989_gcd__mult__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),A3)),C3)) ) ).

% gcd_mult_right
tff(fact_6990_gcd__mult__distrib_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C3)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ).

% gcd_mult_distrib'
tff(fact_6991_normalize__mult,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).

% normalize_mult
tff(fact_6992_coprime__crossproduct,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,D3: A,B2: A,C3: A] :
          ( algebr8660921524188924756oprime(A,A3,D3)
         => ( algebr8660921524188924756oprime(A,B2,C3)
           => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),B2)),aa(A,A,normal6383669964737779283malize(A),D3)) )
            <=> ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
                & ( aa(A,A,normal6383669964737779283malize(A),C3) = aa(A,A,normal6383669964737779283malize(A),D3) ) ) ) ) ) ) ).

% coprime_crossproduct
tff(fact_6993_Lcm__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( ( gcd_Lcm(A,A6) = zero_zero(A) )
          <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),A6)) ) ) ) ).

% Lcm_0_iff
tff(fact_6994_Lcm__nat__infinite,axiom,
    ! [M5: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),M5))
     => ( gcd_Lcm(nat,M5) = zero_zero(nat) ) ) ).

% Lcm_nat_infinite
tff(fact_6995_Lcm__eq__0__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),A6))
         => ( gcd_Lcm(A,A6) = zero_zero(A) ) ) ) ).

% Lcm_eq_0_I
tff(fact_6996_Lcm__0__iff_H,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A6: set(A)] :
          ( ( gcd_Lcm(A,A6) = zero_zero(A) )
        <=> ~ ? [L4: A] :
                ( ( L4 != zero_zero(A) )
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X4),L4)) ) ) ) ) ).

% Lcm_0_iff'
tff(fact_6997_Lcm__no__multiple,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A6: set(A)] :
          ( ! [M: A] :
              ( ( M != zero_zero(A) )
             => ? [X5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A6))
                  & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X5),M)) ) )
         => ( gcd_Lcm(A,A6) = zero_zero(A) ) ) ) ).

% Lcm_no_multiple
tff(fact_6998_normalize__idem__imp__is__unit__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
          <=> ( A3 = one_one(A) ) ) ) ) ).

% normalize_idem_imp_is_unit_iff
tff(fact_6999_is__unit__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) ) ) ) ).

% is_unit_normalize
tff(fact_7000_normalize__1__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).

% normalize_1_iff
tff(fact_7001_associated__unit,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A))) ) ) ) ).

% associated_unit
tff(fact_7002_lcm__gcd__prod,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% lcm_gcd_prod
tff(fact_7003_Gcd__mult,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [C3: A,A6: set(A)] : gcd_Gcd(A,image2(A,A,aa(A,fun(A,A),times_times(A),C3),A6)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),gcd_Gcd(A,A6))) ) ).

% Gcd_mult
tff(fact_7004_lcm__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ) ).

% lcm_coprime
tff(fact_7005_lcm__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ).

% lcm_gcd
tff(fact_7006_Lcm__fin__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(A),B2: A] :
          ( ( A6 != bot_bot(set(A)) )
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),image2(A,A,aa(A,fun(A,A),times_times(A),B2),A6)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(set(A),A,semiring_gcd_Lcm_fin(A),A6))) ) ) ) ).

% Lcm_fin_mult
tff(fact_7007_Gcd__fin__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(A),B2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A6))
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),image2(A,A,aa(A,fun(A,A),times_times(A),B2),A6)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(set(A),A,semiring_gcd_Gcd_fin(A),A6))) ) ) ) ).

% Gcd_fin_mult
tff(fact_7008_Lcm__nat__def,axiom,
    ! [M5: set(nat)] :
      ( ( pp(aa(set(nat),bool,finite_finite2(nat),M5))
       => ( gcd_Lcm(nat,M5) = lattic5214292709420241887eutr_F(nat,gcd_lcm(nat),one_one(nat),M5) ) )
      & ( ~ pp(aa(set(nat),bool,finite_finite2(nat),M5))
       => ( gcd_Lcm(nat,M5) = zero_zero(nat) ) ) ) ).

% Lcm_nat_def
tff(fact_7009_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde6485984586167503788ce_set(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).

% Lcm_fin.bounded_quasi_semilattice_set_axioms
tff(fact_7010_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde6485984586167503788ce_set(A,gcd_gcd(A),zero_zero(A),one_one(A),normal6383669964737779283malize(A)) ) ).

% Gcd_fin.bounded_quasi_semilattice_set_axioms
tff(fact_7011_lcm_Obounded__quasi__semilattice__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde8507323023520639062attice(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).

% lcm.bounded_quasi_semilattice_axioms
tff(fact_7012_gcd_Obounded__quasi__semilattice__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde8507323023520639062attice(A,gcd_gcd(A),zero_zero(A),one_one(A),normal6383669964737779283malize(A)) ) ).

% gcd.bounded_quasi_semilattice_axioms
tff(fact_7013_normalize__div,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : divide_divide(A,aa(A,A,normal6383669964737779283malize(A),A3),A3) = divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,A3)) ) ).

% normalize_div
tff(fact_7014_normalize__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,normal6383669964737779283malize(A),unit_f5069060285200089521factor(A,A3)) = one_one(A) ) ) ) ).

% normalize_unit_factor
tff(fact_7015_unit__factor__simps_I1_J,axiom,
    unit_f5069060285200089521factor(nat,zero_zero(nat)) = zero_zero(nat) ).

% unit_factor_simps(1)
tff(fact_7016_unit__factor__idem,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : unit_f5069060285200089521factor(A,unit_f5069060285200089521factor(A,A3)) = unit_f5069060285200089521factor(A,A3) ) ).

% unit_factor_idem
tff(fact_7017_unit__factor__simps_I2_J,axiom,
    ! [N: nat] : unit_f5069060285200089521factor(nat,aa(nat,nat,suc,N)) = one_one(nat) ).

% unit_factor_simps(2)
tff(fact_7018_unit__factor__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( unit_f5069060285200089521factor(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% unit_factor_eq_0_iff
tff(fact_7019_unit__factor__0,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ( unit_f5069060285200089521factor(A,zero_zero(A)) = zero_zero(A) ) ) ).

% unit_factor_0
tff(fact_7020_unit__factor__1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( unit_f5069060285200089521factor(A,one_one(A)) = one_one(A) ) ) ).

% unit_factor_1
tff(fact_7021_unit__factor__mult__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,A3)),aa(A,A,normal6383669964737779283malize(A),A3)) = A3 ) ).

% unit_factor_mult_normalize
tff(fact_7022_normalize__mult__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),unit_f5069060285200089521factor(A,A3)) = A3 ) ).

% normalize_mult_unit_factor
tff(fact_7023_div__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : divide_divide(A,A3,aa(A,A,normal6383669964737779283malize(A),A3)) = unit_f5069060285200089521factor(A,A3) ) ).

% div_normalize
tff(fact_7024_div__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : divide_divide(A,A3,unit_f5069060285200089521factor(A,A3)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% div_unit_factor
tff(fact_7025_inv__unit__factor__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,A3)) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% inv_unit_factor_eq_0_iff
tff(fact_7026_mult__one__div__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),divide_divide(A,one_one(A),unit_f5069060285200089521factor(A,B2))) = divide_divide(A,A3,unit_f5069060285200089521factor(A,B2)) ) ).

% mult_one_div_unit_factor
tff(fact_7027_unit__factor__mult__unit__left,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),unit_f5069060285200089521factor(A,B2)) ) ) ) ).

% unit_factor_mult_unit_left
tff(fact_7028_unit__factor__mult__unit__right,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,B2)),A3) ) ) ) ).

% unit_factor_mult_unit_right
tff(fact_7029_unit__factor__lcm,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( ( ( A3 = zero_zero(A) )
              | ( B2 = zero_zero(A) ) )
           => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = zero_zero(A) ) )
          & ( ~ ( ( A3 = zero_zero(A) )
                | ( B2 = zero_zero(A) ) )
           => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = one_one(A) ) ) ) ) ).

% unit_factor_lcm
tff(fact_7030_unit__factor__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( unit_f5069060285200089521factor(A,aa(A,A,normal6383669964737779283malize(A),A3)) = one_one(A) ) ) ) ).

% unit_factor_normalize
tff(fact_7031_lcm__mult__distrib,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [K2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),K2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),K2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),K2),B2))),unit_f5069060285200089521factor(A,K2)) ) ).

% lcm_mult_distrib
tff(fact_7032_mult__lcm__right,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))),unit_f5069060285200089521factor(A,C3)) ) ).

% mult_lcm_right
tff(fact_7033_mult__lcm__left,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,C3)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ).

% mult_lcm_left
tff(fact_7034_normalize__unit__factor__eqI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => ( ( unit_f5069060285200089521factor(A,A3) = unit_f5069060285200089521factor(A,B2) )
           => ( A3 = B2 ) ) ) ) ).

% normalize_unit_factor_eqI
tff(fact_7035_dvd__unit__factor__div,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( unit_f5069060285200089521factor(A,divide_divide(A,A3,B2)) = divide_divide(A,unit_f5069060285200089521factor(A,A3),unit_f5069060285200089521factor(A,B2)) ) ) ) ).

% dvd_unit_factor_div
tff(fact_7036_mult__gcd__left,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C3: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,C3)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))) ) ).

% mult_gcd_left
tff(fact_7037_mult__gcd__right,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))),unit_f5069060285200089521factor(A,C3)) ) ).

% mult_gcd_right
tff(fact_7038_gcd__mult__distrib,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [K2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),K2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),K2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),K2),B2))),unit_f5069060285200089521factor(A,K2)) ) ).

% gcd_mult_distrib
tff(fact_7039_is__unit__unit__factor,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( unit_f5069060285200089521factor(A,A3) = A3 ) ) ) ).

% is_unit_unit_factor
tff(fact_7040_unit__factor__nat__def,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( unit_f5069060285200089521factor(nat,N) = zero_zero(nat) ) )
      & ( ( N != zero_zero(nat) )
       => ( unit_f5069060285200089521factor(nat,N) = one_one(nat) ) ) ) ).

% unit_factor_nat_def
tff(fact_7041_unit__factor__dvd,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),unit_f5069060285200089521factor(A,A3)),B2)) ) ) ).

% unit_factor_dvd
tff(fact_7042_unit__factor__mult,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [A3: A,B2: A] : unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,A3)),unit_f5069060285200089521factor(A,B2)) ) ).

% unit_factor_mult
tff(fact_7043_unit__factor__self,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),unit_f5069060285200089521factor(A,A3)),A3)) ) ).

% unit_factor_self
tff(fact_7044_unit__factor__is__unit,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),unit_f5069060285200089521factor(A,A3)),one_one(A))) ) ) ).

% unit_factor_is_unit
tff(fact_7045_unit__factor__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( ( ( A3 = zero_zero(A) )
              & ( B2 = zero_zero(A) ) )
           => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = zero_zero(A) ) )
          & ( ~ ( ( A3 = zero_zero(A) )
                & ( B2 = zero_zero(A) ) )
           => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = one_one(A) ) ) ) ) ).

% unit_factor_gcd
tff(fact_7046_coprime__crossproduct_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [B2: A,D3: A,A3: A,C3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( unit_f5069060285200089521factor(A,B2) = unit_f5069060285200089521factor(A,D3) )
           => ( algebr8660921524188924756oprime(A,A3,B2)
             => ( algebr8660921524188924756oprime(A,C3,D3)
               => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
                <=> ( ( A3 = C3 )
                    & ( B2 = D3 ) ) ) ) ) ) ) ) ).

% coprime_crossproduct'
tff(fact_7047_unit__factor__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A6: set(A)] :
          ( ( ( gcd_Lcm(A,A6) = zero_zero(A) )
           => ( unit_f5069060285200089521factor(A,gcd_Lcm(A,A6)) = zero_zero(A) ) )
          & ( ( gcd_Lcm(A,A6) != zero_zero(A) )
           => ( unit_f5069060285200089521factor(A,gcd_Lcm(A,A6)) = one_one(A) ) ) ) ) ).

% unit_factor_Lcm
tff(fact_7048_unit__factor__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A6: set(A)] :
          ( ( ( gcd_Gcd(A,A6) = zero_zero(A) )
           => ( unit_f5069060285200089521factor(A,gcd_Gcd(A,A6)) = zero_zero(A) ) )
          & ( ( gcd_Gcd(A,A6) != zero_zero(A) )
           => ( unit_f5069060285200089521factor(A,gcd_Gcd(A,A6)) = one_one(A) ) ) ) ) ).

% unit_factor_Gcd
tff(fact_7049_normalize__idem__imp__unit__factor__eq,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
         => ( unit_f5069060285200089521factor(A,A3) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ) ) ).

% normalize_idem_imp_unit_factor_eq
tff(fact_7050_unit__factor__Lcm__fin,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(A)] : unit_f5069060285200089521factor(A,aa(set(A),A,semiring_gcd_Lcm_fin(A),A6)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A6)),zero_zero(A)))) ) ).

% unit_factor_Lcm_fin
tff(fact_7051_unit__factor__Gcd__fin,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A6: set(A)] : unit_f5069060285200089521factor(A,aa(set(A),A,semiring_gcd_Gcd_fin(A),A6)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A6)),zero_zero(A)))) ) ).

% unit_factor_Gcd_fin
tff(fact_7052_map__comp__None__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType,M1: fun(B,option(A)),M22: fun(C,option(B)),K2: C] :
      ( ( map_comp(B,A,C,M1,M22,K2) = none(A) )
    <=> ( ( aa(C,option(B),M22,K2) = none(B) )
        | ? [K10: B] :
            ( ( aa(C,option(B),M22,K2) = aa(B,option(B),some(B),K10) )
            & ( aa(B,option(A),M1,K10) = none(A) ) ) ) ) ).

% map_comp_None_iff
tff(fact_7053_map__le__imp__upd__le,axiom,
    ! [A: $tType,B: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),X: A,Y: B] :
      ( map_le(A,B,M1,M22)
     => map_le(A,B,fun_upd(A,option(B),M1,X,none(B)),fun_upd(A,option(B),M22,X,aa(B,option(B),some(B),Y))) ) ).

% map_le_imp_upd_le
tff(fact_7054_map__comp__simps_I2_J,axiom,
    ! [B: $tType,C: $tType,A: $tType,M22: fun(B,option(A)),K2: B,K7: A,M1: fun(A,option(C))] :
      ( ( aa(B,option(A),M22,K2) = aa(A,option(A),some(A),K7) )
     => ( map_comp(A,C,B,M1,M22,K2) = aa(A,option(C),M1,K7) ) ) ).

% map_comp_simps(2)
tff(fact_7055_map__comp__Some__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType,M1: fun(B,option(A)),M22: fun(C,option(B)),K2: C,V3: A] :
      ( ( map_comp(B,A,C,M1,M22,K2) = aa(A,option(A),some(A),V3) )
    <=> ? [K10: B] :
          ( ( aa(C,option(B),M22,K2) = aa(B,option(B),some(B),K10) )
          & ( aa(B,option(A),M1,K10) = aa(A,option(A),some(A),V3) ) ) ) ).

% map_comp_Some_iff
tff(fact_7056_is__num_Ocases,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A] :
          ( neg_numeral_is_num(A,A3)
         => ( ( A3 != one_one(A) )
           => ( ! [X3: A] :
                  ( ( A3 = aa(A,A,uminus_uminus(A),X3) )
                 => ~ neg_numeral_is_num(A,X3) )
             => ~ ! [X3: A,Y3: A] :
                    ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y3) )
                   => ( neg_numeral_is_num(A,X3)
                     => ~ neg_numeral_is_num(A,Y3) ) ) ) ) ) ) ).

% is_num.cases
tff(fact_7057_is__num_Osimps,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A] :
          ( neg_numeral_is_num(A,A3)
        <=> ( ( A3 = one_one(A) )
            | ? [X4: A] :
                ( ( A3 = aa(A,A,uminus_uminus(A),X4) )
                & neg_numeral_is_num(A,X4) )
            | ? [X4: A,Y5: A] :
                ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y5) )
                & neg_numeral_is_num(A,X4)
                & neg_numeral_is_num(A,Y5) ) ) ) ) ).

% is_num.simps
tff(fact_7058_is__num__normalize_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] :
          ( neg_numeral_is_num(A,X)
         => neg_numeral_is_num(A,aa(A,A,uminus_uminus(A),X)) ) ) ).

% is_num_normalize(5)
tff(fact_7059_is__num__normalize_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => neg_numeral_is_num(A,one_one(A)) ) ).

% is_num_normalize(4)
tff(fact_7060_is__num__normalize_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A,Y: A] :
          ( neg_numeral_is_num(A,X)
         => ( neg_numeral_is_num(A,Y)
           => neg_numeral_is_num(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) ) ) ) ).

% is_num_normalize(6)
tff(fact_7061_is__num__add__commute,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A,Y: A] :
          ( neg_numeral_is_num(A,X)
         => ( neg_numeral_is_num(A,Y)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),X) ) ) ) ) ).

% is_num_add_commute
tff(fact_7062_is__num__add__left__commute,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A,Y: A,Z2: A] :
          ( neg_numeral_is_num(A,X)
         => ( neg_numeral_is_num(A,Y)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2)) ) ) ) ) ).

% is_num_add_left_commute
tff(fact_7063_is__num__numeral,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_is_num(A,aa(num,A,numeral_numeral(A),K2)) ) ).

% is_num_numeral
tff(fact_7064_times__num__def,axiom,
    ! [M2: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M2),N) = num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,nat_of_num,M2)),aa(num,nat,nat_of_num,N))) ).

% times_num_def
tff(fact_7065_arg__max__nat__lemma,axiom,
    ! [A: $tType,P2: fun(A,bool),K2: A,F3: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P2,K2))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y3)),B2)) )
       => ( pp(aa(A,bool,P2,lattices_ord_arg_max(A,nat,F3,P2)))
          & ! [Y4: A] :
              ( pp(aa(A,bool,P2,Y4))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,Y4)),aa(A,nat,F3,lattices_ord_arg_max(A,nat,F3,P2)))) ) ) ) ) ).

% arg_max_nat_lemma
tff(fact_7066_nat__of__num__code_I2_J,axiom,
    ! [N: num] : aa(num,nat,nat_of_num,bit0(N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,N)),aa(num,nat,nat_of_num,N)) ).

% nat_of_num_code(2)
tff(fact_7067_less__eq__num__def,axiom,
    ! [M2: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M2),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,nat_of_num,M2)),aa(num,nat,nat_of_num,N))) ) ).

% less_eq_num_def
tff(fact_7068_less__num__def,axiom,
    ! [M2: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M2),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,nat_of_num,M2)),aa(num,nat,nat_of_num,N))) ) ).

% less_num_def
tff(fact_7069_nat__of__num__inc,axiom,
    ! [X: num] : aa(num,nat,nat_of_num,inc(X)) = aa(nat,nat,suc,aa(num,nat,nat_of_num,X)) ).

% nat_of_num_inc
tff(fact_7070_nat__of__num__mult,axiom,
    ! [X: num,Y: num] : aa(num,nat,nat_of_num,aa(num,num,aa(num,fun(num,num),times_times(num),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,Y)) ).

% nat_of_num_mult
tff(fact_7071_nat__of__num__numeral,axiom,
    nat_of_num = numeral_numeral(nat) ).

% nat_of_num_numeral
tff(fact_7072_num__eq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
    <=> ( aa(num,nat,nat_of_num,X) = aa(num,nat,nat_of_num,Y) ) ) ).

% num_eq_iff
tff(fact_7073_nat__of__num__inverse,axiom,
    ! [X: num] : num_of_nat(aa(num,nat,nat_of_num,X)) = X ).

% nat_of_num_inverse
tff(fact_7074_nat__of__num_Osimps_I2_J,axiom,
    ! [X: num] : aa(num,nat,nat_of_num,bit0(X)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,X)) ).

% nat_of_num.simps(2)
tff(fact_7075_nat__of__num__pos,axiom,
    ! [X: num] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,nat_of_num,X))) ).

% nat_of_num_pos
tff(fact_7076_nat__of__num__neq__0,axiom,
    ! [X: num] : aa(num,nat,nat_of_num,X) != zero_zero(nat) ).

% nat_of_num_neq_0
tff(fact_7077_nat__of__num__code_I1_J,axiom,
    aa(num,nat,nat_of_num,one2) = one_one(nat) ).

% nat_of_num_code(1)
tff(fact_7078_nat__of__num__add,axiom,
    ! [X: num,Y: num] : aa(num,nat,nat_of_num,aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,Y)) ).

% nat_of_num_add
tff(fact_7079_nat__of__num__sqr,axiom,
    ! [X: num] : aa(num,nat,nat_of_num,sqr(X)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,X)) ).

% nat_of_num_sqr
tff(fact_7080_arg__max__equality,axiom,
    ! [A: $tType,C: $tType] :
      ( order(A)
     => ! [P2: fun(C,bool),K2: C,F3: fun(C,A)] :
          ( pp(aa(C,bool,P2,K2))
         => ( ! [X3: C] :
                ( pp(aa(C,bool,P2,X3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F3,X3)),aa(C,A,F3,K2))) )
           => ( aa(C,A,F3,lattices_ord_arg_max(C,A,F3,P2)) = aa(C,A,F3,K2) ) ) ) ) ).

% arg_max_equality
tff(fact_7081_nat__of__num_Osimps_I1_J,axiom,
    aa(num,nat,nat_of_num,one2) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_of_num.simps(1)
tff(fact_7082_nat__of__num_Osimps_I3_J,axiom,
    ! [X: num] : aa(num,nat,nat_of_num,aa(num,num,bit1,X)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,X))) ).

% nat_of_num.simps(3)
tff(fact_7083_num__of__nat__inverse,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(num,nat,nat_of_num,num_of_nat(N)) = N ) ) ).

% num_of_nat_inverse
tff(fact_7084_nat__of__num__code_I3_J,axiom,
    ! [N: num] : aa(num,nat,nat_of_num,aa(num,num,bit1,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,N)),aa(num,nat,nat_of_num,N))) ).

% nat_of_num_code(3)
tff(fact_7085_plus__num__def,axiom,
    ! [M2: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),M2),N) = num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,M2)),aa(num,nat,nat_of_num,N))) ).

% plus_num_def
tff(fact_7086_arg__max__nat__le,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A,F3: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P2,X))
     => ( ! [Y3: A] :
            ( pp(aa(A,bool,P2,Y3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y3)),B2)) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X)),aa(A,nat,F3,lattices_ord_arg_max(A,nat,F3,P2)))) ) ) ).

% arg_max_nat_le
tff(fact_7087_trans__join,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( trans(A,R)
    <=> ! [X4: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),X4),R))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_ahz(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X4)) ) ) ).

% trans_join
tff(fact_7088_Gr__def,axiom,
    ! [B: $tType,A: $tType,A6: set(A),F3: fun(A,B)] : bNF_Gr(A,B,A6,F3) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_aia(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),A6),F3)) ).

% Gr_def
tff(fact_7089_trans__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( trans(A,R)
    <=> ! [X4: A,Y5: A,Z3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5)),R))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z3)),R))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Z3)),R)) ) ) ) ).

% trans_def
tff(fact_7090_transI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [X3: A,Y3: A,Z: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z)),R)) ) )
     => trans(A,R) ) ).

% transI
tff(fact_7091_transE,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
      ( trans(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
       => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R))
         => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),R)) ) ) ) ).

% transE
tff(fact_7092_transD,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
      ( trans(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
       => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R))
         => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),R)) ) ) ) ).

% transD
tff(fact_7093_GrD1,axiom,
    ! [B: $tType,A: $tType,X: A,Fx: B,A6: set(A),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx)),bNF_Gr(A,B,A6,F3)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6)) ) ).

% GrD1
tff(fact_7094_GrD2,axiom,
    ! [A: $tType,B: $tType,X: A,Fx: B,A6: set(A),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Fx)),bNF_Gr(A,B,A6,F3)))
     => ( aa(A,B,F3,X) = Fx ) ) ).

% GrD2
tff(fact_7095_natLeq__trans,axiom,
    trans(nat,bNF_Ca8665028551170535155natLeq) ).

% natLeq_trans
tff(fact_7096_transp__trans__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( transp(A,aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
    <=> trans(A,R) ) ).

% transp_trans_eq
tff(fact_7097_trans__singleton,axiom,
    ! [A: $tType,A3: A] : trans(A,aa(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),A3),A3)),bot_bot(set(product_prod(A,A))))) ).

% trans_singleton
tff(fact_7098_wf__finite__segments,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irrefl(A,R)
     => ( trans(A,R)
       => ( ! [X3: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aib(set(product_prod(A,A)),fun(A,fun(A,bool)),R),X3))))
         => wf(A,R) ) ) ) ).

% wf_finite_segments
tff(fact_7099_underS__incr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( trans(A,R)
     => ( antisym(A,R)
       => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R,A3)),order_underS(A,R,B2))) ) ) ) ).

% underS_incr
tff(fact_7100_natLeq__antisym,axiom,
    antisym(nat,bNF_Ca8665028551170535155natLeq) ).

% natLeq_antisym
tff(fact_7101_irreflI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [A5: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5)),R2))
     => irrefl(A,R2) ) ).

% irreflI
tff(fact_7102_irrefl__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irrefl(A,R)
    <=> ! [A7: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),A7)),R)) ) ).

% irrefl_def
tff(fact_7103_antisym__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( antisym(A,R)
    <=> ! [X4: A,Y5: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5)),R))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X4)),R))
           => ( X4 = Y5 ) ) ) ) ).

% antisym_def
tff(fact_7104_antisymI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [X3: A,Y3: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3)),R))
           => ( X3 = Y3 ) ) )
     => antisym(A,R) ) ).

% antisymI
tff(fact_7105_antisymD,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( antisym(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R))
         => ( A3 = B2 ) ) ) ) ).

% antisymD
tff(fact_7106_irreflp__irrefl__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irreflp(A,aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R2))
    <=> irrefl(A,R2) ) ).

% irreflp_irrefl_eq
tff(fact_7107_antisymp__antisym__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( antisymp(A,aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
    <=> antisym(A,R) ) ).

% antisymp_antisym_eq
tff(fact_7108_prod_H__def,axiom,
    ! [C: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ( groups1962203154675924110t_prod(C,A) = groups_comm_monoid_G(A,C,times_times(A),one_one(A)) ) ) ).

% prod'_def
tff(fact_7109_prod_Ocomm__monoid__list__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => groups4802862169904069756st_set(A,times_times(A),one_one(A)) ) ).

% prod.comm_monoid_list_set_axioms
tff(fact_7110_sum_H__def,axiom,
    ! [C: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ( groups1027152243600224163dd_sum(C,A) = groups_comm_monoid_G(A,C,plus_plus(A),zero_zero(A)) ) ) ).

% sum'_def
tff(fact_7111_sum_Ocomm__monoid__list__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => groups4802862169904069756st_set(A,plus_plus(A),zero_zero(A)) ) ).

% sum.comm_monoid_list_set_axioms
tff(fact_7112_VEBT__internal_Ooption__comp__shift_Opelims_I3_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: option(A),Xb: option(A)] :
      ( ~ vEBT_V6923181176774028177_shift(A,X,Xa2,Xb)
     => ( accp(product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),vEBT_V4810408830578336424ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),Xa2),Xb)))
       => ( ( ( Xa2 = none(A) )
           => ~ accp(product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),vEBT_V4810408830578336424ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Xb))) )
         => ( ! [V2: A] :
                ( ( Xa2 = aa(A,option(A),some(A),V2) )
               => ( ( Xb = none(A) )
                 => ~ accp(product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),vEBT_V4810408830578336424ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V2)),none(A)))) ) )
           => ~ ! [X3: A] :
                  ( ( Xa2 = aa(A,option(A),some(A),X3) )
                 => ! [Y3: A] :
                      ( ( Xb = aa(A,option(A),some(A),Y3) )
                     => ( accp(product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),vEBT_V4810408830578336424ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),X3)),aa(A,option(A),some(A),Y3))))
                       => pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3)) ) ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.pelims(3)
tff(fact_7113_VEBT__internal_Ooption__comp__shift_Opelims_I1_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: option(A),Xb: option(A),Y: bool] :
      ( ( vEBT_V6923181176774028177_shift(A,X,Xa2,Xb)
      <=> pp(Y) )
     => ( accp(product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),vEBT_V4810408830578336424ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),Xa2),Xb)))
       => ( ( ( Xa2 = none(A) )
           => ( ~ pp(Y)
             => ~ accp(product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),vEBT_V4810408830578336424ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Xb))) ) )
         => ( ! [V2: A] :
                ( ( Xa2 = aa(A,option(A),some(A),V2) )
               => ( ( Xb = none(A) )
                 => ( ~ pp(Y)
                   => ~ accp(product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),vEBT_V4810408830578336424ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V2)),none(A)))) ) ) )
           => ~ ! [X3: A] :
                  ( ( Xa2 = aa(A,option(A),some(A),X3) )
                 => ! [Y3: A] :
                      ( ( Xb = aa(A,option(A),some(A),Y3) )
                     => ( ( pp(Y)
                        <=> pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3)) )
                       => ~ accp(product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),vEBT_V4810408830578336424ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),X3)),aa(A,option(A),some(A),Y3)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.pelims(1)
tff(fact_7114_VEBT__internal_Ooption__comp__shift_Oelims_I2_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: option(A),Xb: option(A)] :
      ( vEBT_V6923181176774028177_shift(A,X,Xa2,Xb)
     => ~ ! [X3: A] :
            ( ( Xa2 = aa(A,option(A),some(A),X3) )
           => ! [Y3: A] :
                ( ( Xb = aa(A,option(A),some(A),Y3) )
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3)) ) ) ) ).

% VEBT_internal.option_comp_shift.elims(2)
tff(fact_7115_VEBT__internal_Ooption__comp__shift_Osimps_I3_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,bool)),X: A,Y: A] :
      ( vEBT_V6923181176774028177_shift(A,F3,aa(A,option(A),some(A),X),aa(A,option(A),some(A),Y))
    <=> pp(aa(A,bool,aa(A,fun(A,bool),F3,X),Y)) ) ).

% VEBT_internal.option_comp_shift.simps(3)
tff(fact_7116_VEBT__internal_Olesseq_Oelims_I3_J,axiom,
    ! [X: option(nat),Xa2: option(nat)] :
      ( ~ vEBT_VEBT_lesseq(X,Xa2)
     => ~ vEBT_V6923181176774028177_shift(nat,ord_less_eq(nat),X,Xa2) ) ).

% VEBT_internal.lesseq.elims(3)
tff(fact_7117_VEBT__internal_Olesseq_Oelims_I2_J,axiom,
    ! [X: option(nat),Xa2: option(nat)] :
      ( vEBT_VEBT_lesseq(X,Xa2)
     => vEBT_V6923181176774028177_shift(nat,ord_less_eq(nat),X,Xa2) ) ).

% VEBT_internal.lesseq.elims(2)
tff(fact_7118_VEBT__internal_Olesseq_Oelims_I1_J,axiom,
    ! [X: option(nat),Xa2: option(nat),Y: bool] :
      ( ( vEBT_VEBT_lesseq(X,Xa2)
      <=> pp(Y) )
     => ( pp(Y)
      <=> vEBT_V6923181176774028177_shift(nat,ord_less_eq(nat),X,Xa2) ) ) ).

% VEBT_internal.lesseq.elims(1)
tff(fact_7119_VEBT__internal_Olesseq_Osimps,axiom,
    ! [X: option(nat),Y: option(nat)] :
      ( vEBT_VEBT_lesseq(X,Y)
    <=> vEBT_V6923181176774028177_shift(nat,ord_less_eq(nat),X,Y) ) ).

% VEBT_internal.lesseq.simps
tff(fact_7120_VEBT__internal_Ooption__comp__shift_Opelims_I2_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: option(A),Xb: option(A)] :
      ( vEBT_V6923181176774028177_shift(A,X,Xa2,Xb)
     => ( accp(product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),vEBT_V4810408830578336424ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),Xa2),Xb)))
       => ~ ! [X3: A] :
              ( ( Xa2 = aa(A,option(A),some(A),X3) )
             => ! [Y3: A] :
                  ( ( Xb = aa(A,option(A),some(A),Y3) )
                 => ( accp(product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),vEBT_V4810408830578336424ft_rel(A),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A))),aa(fun(A,fun(A,bool)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,bool)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(option(A),option(A))),X),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),X3)),aa(A,option(A),some(A),Y3))))
                   => ~ pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3)) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.pelims(2)
tff(fact_7121_VEBT__internal_Ooption__comp__shift_Oelims_I3_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: option(A),Xb: option(A)] :
      ( ~ vEBT_V6923181176774028177_shift(A,X,Xa2,Xb)
     => ( ( Xa2 != none(A) )
       => ( ( ? [V2: A] : Xa2 = aa(A,option(A),some(A),V2)
           => ( Xb != none(A) ) )
         => ~ ! [X3: A] :
                ( ( Xa2 = aa(A,option(A),some(A),X3) )
               => ! [Y3: A] :
                    ( ( Xb = aa(A,option(A),some(A),Y3) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3)) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.elims(3)
tff(fact_7122_VEBT__internal_Ooption__comp__shift_Oelims_I1_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa2: option(A),Xb: option(A),Y: bool] :
      ( ( vEBT_V6923181176774028177_shift(A,X,Xa2,Xb)
      <=> pp(Y) )
     => ( ( ( Xa2 = none(A) )
         => pp(Y) )
       => ( ( ? [V2: A] : Xa2 = aa(A,option(A),some(A),V2)
           => ( ( Xb = none(A) )
             => pp(Y) ) )
         => ~ ! [X3: A] :
                ( ( Xa2 = aa(A,option(A),some(A),X3) )
               => ! [Y3: A] :
                    ( ( Xb = aa(A,option(A),some(A),Y3) )
                   => ( pp(Y)
                    <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Y3)) ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.elims(1)
tff(fact_7123_VEBT__internal_Ooption__comp__shift_Osimps_I2_J,axiom,
    ! [A: $tType,Uw2: fun(A,fun(A,bool)),V3: A] : ~ vEBT_V6923181176774028177_shift(A,Uw2,aa(A,option(A),some(A),V3),none(A)) ).

% VEBT_internal.option_comp_shift.simps(2)
tff(fact_7124_fold__atLeastAtMost__nat,axiom,
    ! [A: $tType,F3: fun(nat,fun(A,A)),A3: nat,B2: nat,Acc2: A] :
      ( finite6289374366891150609ommute(nat,A,F3)
     => ( set_fo6178422350223883121st_nat(A,F3,A3,B2,Acc2) = finite_fold(nat,A,F3,Acc2,set_or1337092689740270186AtMost(nat,A3,B2)) ) ) ).

% fold_atLeastAtMost_nat
tff(fact_7125_max_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semilattice_order(A,ord_max(A),aTP_Lamp_lw(A,fun(A,bool)),aTP_Lamp_aen(A,fun(A,bool))) ) ).

% max.semilattice_order_axioms
tff(fact_7126_inf_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => semilattice_order(A,inf_inf(A),ord_less_eq(A),ord_less(A)) ) ).

% inf.semilattice_order_axioms
tff(fact_7127_min_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semilattice_order(A,ord_min(A),ord_less_eq(A),ord_less(A)) ) ).

% min.semilattice_order_axioms
tff(fact_7128_sup_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => semilattice_order(A,sup_sup(A),aTP_Lamp_aeo(A,fun(A,bool)),aTP_Lamp_aep(A,fun(A,bool))) ) ).

% sup.semilattice_order_axioms
tff(fact_7129_under__incr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( trans(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_under(A,R,A3)),order_under(A,R,B2))) ) ) ).

% under_incr
tff(fact_7130_bdd__below__primitive__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ( condit1013018076250108175_below(A) = condit16957441358409770ng_bdd(A,aTP_Lamp_aeh(A,fun(A,bool))) ) ) ).

% bdd_below_primitive_def
tff(fact_7131_under__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_under(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aib(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).

% under_def
tff(fact_7132_bdd__above__primitive__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ( condit941137186595557371_above(A) = condit16957441358409770ng_bdd(A,ord_less_eq(A)) ) ) ).

% bdd_above_primitive_def
tff(fact_7133_has__vector__derivative__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(real,A),F6: A,X: real,S2: set(real),G3: fun(real,A),G5: A] :
          ( has_ve8173657378732805170vative(A,F3,F6,topolo174197925503356063within(real,X,S2))
         => ( has_ve8173657378732805170vative(A,G3,G5,topolo174197925503356063within(real,X,S2))
           => has_ve8173657378732805170vative(A,aa(fun(real,A),fun(real,A),aTP_Lamp_aic(fun(real,A),fun(fun(real,A),fun(real,A)),F3),G3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,F3,X)),G5)),aa(A,A,aa(A,fun(A,A),times_times(A),F6),aa(real,A,G3,X))),topolo174197925503356063within(real,X,S2)) ) ) ) ).

% has_vector_derivative_mult
tff(fact_7134_integer__of__nat__0,axiom,
    code_integer_of_nat(zero_zero(nat)) = zero_zero(code_integer) ).

% integer_of_nat_0
tff(fact_7135_has__vector__derivative__mult__right,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(real,A),X: A,F4: filter(real),A3: A] :
          ( has_ve8173657378732805170vative(A,F3,X,F4)
         => has_ve8173657378732805170vative(A,aa(A,fun(real,A),aTP_Lamp_aid(fun(real,A),fun(A,fun(real,A)),F3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),X),F4) ) ) ).

% has_vector_derivative_mult_right
tff(fact_7136_has__vector__derivative__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(real,A),X: A,F4: filter(real),A3: A] :
          ( has_ve8173657378732805170vative(A,F3,X,F4)
         => has_ve8173657378732805170vative(A,aa(A,fun(real,A),aTP_Lamp_aie(fun(real,A),fun(A,fun(real,A)),F3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),X),A3),F4) ) ) ).

% has_vector_derivative_mult_left
tff(fact_7137_has__vector__derivative__const,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C3: A,Net: filter(real)] : has_ve8173657378732805170vative(A,aTP_Lamp_aif(A,fun(real,A),C3),zero_zero(A),Net) ) ).

% has_vector_derivative_const
tff(fact_7138_equiv__class__nondisjoint,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X: A,A3: A,B2: A] :
      ( equiv_equiv(A,A6,R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(A,A,R,aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))),image(A,A,R,aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ).

% equiv_class_nondisjoint
tff(fact_7139_bdd__below_Opreordering__bdd__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit622319405099724424ng_bdd(A,aTP_Lamp_aeh(A,fun(A,bool)),aTP_Lamp_aig(A,fun(A,bool))) ) ).

% bdd_below.preordering_bdd_axioms
tff(fact_7140_equiv__class__eq__iff,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A6,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
      <=> ( ( image(A,A,R,aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = image(A,A,R,aa(set(A),set(A),insert(A,Y),bot_bot(set(A)))) )
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A6)) ) ) ) ).

% equiv_class_eq_iff
tff(fact_7141_eq__equiv__class__iff,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A6,R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A6))
         => ( ( image(A,A,R,aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = image(A,A,R,aa(set(A),set(A),insert(A,Y),bot_bot(set(A)))) )
          <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R)) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_7142_equiv__class__eq,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A6,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => ( image(A,A,R,aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))) = image(A,A,R,aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) ) ) ) ).

% equiv_class_eq
tff(fact_7143_eq__equiv__class,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,A6: set(A)] :
      ( ( image(A,A,R,aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))) = image(A,A,R,aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) )
     => ( equiv_equiv(A,A6,R)
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A6))
         => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ) ).

% eq_equiv_class
tff(fact_7144_bdd__above_Opreordering__bdd__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit622319405099724424ng_bdd(A,ord_less_eq(A),ord_less(A)) ) ).

% bdd_above.preordering_bdd_axioms
tff(fact_7145_subset__equiv__class,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),B2: A,A3: A] :
      ( equiv_equiv(A,A6,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R,aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))),image(A,A,R,aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A6))
         => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ) ).

% subset_equiv_class
tff(fact_7146_equiv__class__subset,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A6,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R,aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))),image(A,A,R,aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))) ) ) ).

% equiv_class_subset
tff(fact_7147_proj__iff,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A6,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,X),aa(set(A),set(A),insert(A,Y),bot_bot(set(A))))),A6))
       => ( ( equiv_proj(A,A,R,X) = equiv_proj(A,A,R,Y) )
        <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R)) ) ) ) ).

% proj_iff
tff(fact_7148_finite__refines__card__le,axiom,
    ! [A: $tType,A6: set(A),R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A6,R2)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R2),S3))
       => ( equiv_equiv(A,A6,R2)
         => ( equiv_equiv(A,A6,S3)
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A6,S3))),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A6,R2)))) ) ) ) ) ).

% finite_refines_card_le
tff(fact_7149_in__quotient__imp__closed,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A6,R)
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A6,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),X6)) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_7150_quotient__eq__iff,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X6: set(A),Y6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A6,R)
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A6,R)))
       => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y6),equiv_quotient(A,A6,R)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),Y6))
             => ( ( X6 = Y6 )
              <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R)) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_7151_quotient__eqI,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X6: set(A),Y6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A6,R)
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A6,R)))
       => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y6),equiv_quotient(A,A6,R)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),Y6))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
               => ( X6 = Y6 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_7152_eq__equiv__class__iff2,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X: A,Y: A] :
      ( equiv_equiv(A,A6,R)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A6))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A6))
         => ( ( equiv_quotient(A,aa(set(A),set(A),insert(A,X),bot_bot(set(A))),R) = equiv_quotient(A,aa(set(A),set(A),insert(A,Y),bot_bot(set(A))),R) )
          <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R)) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_7153_in__quotient__imp__in__rel,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X6: set(A),X: A,Y: A] :
      ( equiv_equiv(A,A6,R)
     => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A6,R)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,X),aa(set(A),set(A),insert(A,Y),bot_bot(set(A))))),X6))
         => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R)) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_7154_UN__equiv__class__inject,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R: set(product_prod(A,A)),F3: fun(A,set(B)),X6: set(A),Y6: set(A)] :
      ( equiv_equiv(A,A6,R)
     => ( equiv_congruent(A,set(B),R,F3)
       => ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),image2(A,set(B),F3,X6)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),image2(A,set(B),F3,Y6)) )
         => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),equiv_quotient(A,A6,R)))
           => ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y6),equiv_quotient(A,A6,R)))
             => ( ! [X3: A,Y3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A6))
                   => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A6))
                     => ( ( aa(A,set(B),F3,X3) = aa(A,set(B),F3,Y3) )
                       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y3)),R)) ) ) )
               => ( X6 = Y6 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
tff(fact_7155_disjnt__equiv__class,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A6,R)
     => ( disjnt(A,image(A,A,R,aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))),image(A,A,R,aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))
      <=> ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ).

% disjnt_equiv_class
tff(fact_7156_disjnt__Times1__iff,axiom,
    ! [A: $tType,B: $tType,C5: set(A),A6: set(B),B6: set(B)] :
      ( disjnt(product_prod(A,B),product_Sigma(A,B,C5,aTP_Lamp_aci(set(B),fun(A,set(B)),A6)),product_Sigma(A,B,C5,aTP_Lamp_aci(set(B),fun(A,set(B)),B6)))
    <=> ( ( C5 = bot_bot(set(A)) )
        | disjnt(B,A6,B6) ) ) ).

% disjnt_Times1_iff
tff(fact_7157_disjnt__Times2__iff,axiom,
    ! [B: $tType,A: $tType,A6: set(A),C5: set(B),B6: set(A)] :
      ( disjnt(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),C5)),product_Sigma(A,B,B6,aTP_Lamp_aci(set(B),fun(A,set(B)),C5)))
    <=> ( ( C5 = bot_bot(set(B)) )
        | disjnt(A,A6,B6) ) ) ).

% disjnt_Times2_iff
tff(fact_7158_disjnt__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A6: set(A),C5: fun(A,set(B)),B6: set(A)] :
      ( disjnt(product_prod(A,B),product_Sigma(A,B,A6,C5),product_Sigma(A,B,B6,C5))
    <=> ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6)))
           => ( aa(A,set(B),C5,X4) = bot_bot(set(B)) ) )
        | disjnt(A,A6,B6) ) ) ).

% disjnt_Sigma_iff
tff(fact_7159_congruentD,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F3: fun(A,B),Y: A,Z2: A] :
      ( equiv_congruent(A,B,R,F3)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R))
       => ( aa(A,B,F3,Y) = aa(A,B,F3,Z2) ) ) ) ).

% congruentD
tff(fact_7160_congruentI,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F3: fun(A,B)] :
      ( ! [Y3: A,Z: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R))
         => ( aa(A,B,F3,Y3) = aa(A,B,F3,Z) ) )
     => equiv_congruent(A,B,R,F3) ) ).

% congruentI
tff(fact_7161_congruent2__commuteI,axiom,
    ! [B: $tType,A: $tType,A6: set(A),R: set(product_prod(A,A)),F3: fun(A,fun(A,B))] :
      ( equiv_equiv(A,A6,R)
     => ( ! [Y3: A,Z: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),A6))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),A6))
             => ( aa(A,B,aa(A,fun(A,B),F3,Y3),Z) = aa(A,B,aa(A,fun(A,B),F3,Z),Y3) ) ) )
       => ( ! [Y3: A,Z: A,W: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),W),A6))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R))
               => ( aa(A,B,aa(A,fun(A,B),F3,W),Y3) = aa(A,B,aa(A,fun(A,B),F3,W),Z) ) ) )
         => equiv_congruent2(A,A,B,R,R,F3) ) ) ) ).

% congruent2_commuteI
tff(fact_7162_congruent2I,axiom,
    ! [C: $tType,B: $tType,A: $tType,A16: set(A),R1: set(product_prod(A,A)),A25: set(B),R22: set(product_prod(B,B)),F3: fun(A,fun(B,C))] :
      ( equiv_equiv(A,A16,R1)
     => ( equiv_equiv(B,A25,R22)
       => ( ! [Y3: A,Z: A,W: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),W),A25))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R1))
               => ( aa(B,C,aa(A,fun(B,C),F3,Y3),W) = aa(B,C,aa(A,fun(B,C),F3,Z),W) ) ) )
         => ( ! [Y3: B,Z: B,W: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),W),A16))
               => ( pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y3),Z)),R22))
                 => ( aa(B,C,aa(A,fun(B,C),F3,W),Y3) = aa(B,C,aa(A,fun(B,C),F3,W),Z) ) ) )
           => equiv_congruent2(A,B,C,R1,R22,F3) ) ) ) ) ).

% congruent2I
tff(fact_7163_congruent2I_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F3: fun(A,fun(B,C))] :
      ( ! [Y15: A,Z12: A,Y23: B,Z23: B] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y15),Z12)),R1))
         => ( pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y23),Z23)),R22))
           => ( aa(B,C,aa(A,fun(B,C),F3,Y15),Y23) = aa(B,C,aa(A,fun(B,C),F3,Z12),Z23) ) ) )
     => equiv_congruent2(A,B,C,R1,R22,F3) ) ).

% congruent2I'
tff(fact_7164_congruent2D,axiom,
    ! [A: $tType,C: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F3: fun(A,fun(B,C)),Y1: A,Z1: A,Y2: B,Z22: B] :
      ( equiv_congruent2(A,B,C,R1,R22,F3)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y1),Z1)),R1))
       => ( pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Z22)),R22))
         => ( aa(B,C,aa(A,fun(B,C),F3,Y1),Y2) = aa(B,C,aa(A,fun(B,C),F3,Z1),Z22) ) ) ) ) ).

% congruent2D
tff(fact_7165_equivp__equiv,axiom,
    ! [A: $tType,A6: set(product_prod(A,A))] :
      ( equiv_equiv(A,top_top(set(A)),A6)
    <=> equiv_equivp(A,aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),A6)) ) ).

% equivp_equiv
tff(fact_7166_fstOp__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,P2: fun(A,fun(B,bool)),Q: fun(B,fun(C,bool)),Ac: product_prod(A,C)] : bNF_fstOp(A,B,C,P2,Q,Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Ac)),bNF_pick_middlep(A,B,C,P2,Q,aa(product_prod(A,C),A,product_fst(A,C),Ac),aa(product_prod(A,C),C,product_snd(A,C),Ac))) ).

% fstOp_def
tff(fact_7167_sndOp__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: fun(C,fun(A,bool)),Q: fun(A,fun(B,bool)),Ac: product_prod(C,B)] : bNF_sndOp(C,A,B,P2,Q,Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),bNF_pick_middlep(C,A,B,P2,Q,aa(product_prod(C,B),C,product_fst(C,B),Ac),aa(product_prod(C,B),B,product_snd(C,B),Ac))),aa(product_prod(C,B),B,product_snd(C,B),Ac)) ).

% sndOp_def
tff(fact_7168_reflp__refl__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( reflp(A,aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
    <=> refl_on(A,top_top(set(A)),R) ) ).

% reflp_refl_eq
tff(fact_7169_option_Orel__reflp,axiom,
    ! [A: $tType,R2: fun(A,fun(A,bool))] :
      ( reflp(A,R2)
     => reflp(option(A),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),R2)) ) ).

% option.rel_reflp
tff(fact_7170_card_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_aas(A,fun(nat,nat))) ).

% card.folding_on_axioms
tff(fact_7171_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_zero(code_natural) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(code_natural,nat,code_nat_of_natural,N)),aa(nat,nat,suc,zero_zero(nat)))),aa(code_natural,nat,code_nat_of_natural,N))) ) ).

% natural_decr
tff(fact_7172_times__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),X),Xa2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2)) ).

% times_natural.rep_eq
tff(fact_7173_zero__natural_Orep__eq,axiom,
    aa(code_natural,nat,code_nat_of_natural,zero_zero(code_natural)) = zero_zero(nat) ).

% zero_natural.rep_eq
tff(fact_7174_less__eq__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),Xa2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa2))) ) ).

% less_eq_natural.rep_eq
tff(fact_7175_iterate_Oelims,axiom,
    ! [A: $tType,B: $tType,X: code_natural,Xa2: fun(B,fun(A,product_prod(B,A))),Xb: B,Y: fun(A,product_prod(B,A))] :
      ( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa2),Xb) = Y )
     => ( ( ( X = zero_zero(code_natural) )
         => ( Y = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb) ) )
        & ( ( X != zero_zero(code_natural) )
         => ( Y = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa2,Xb),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa2)) ) ) ) ) ).

% iterate.elims
tff(fact_7176_iterate_Osimps,axiom,
    ! [A: $tType,B: $tType,K2: code_natural,F3: fun(B,fun(A,product_prod(B,A))),X: B] :
      ( ( ( K2 = zero_zero(code_natural) )
       => ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K2,F3),X) = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X) ) )
      & ( ( K2 != zero_zero(code_natural) )
       => ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K2,F3),X) = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F3,X),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K2),one_one(code_natural)),F3)) ) ) ) ).

% iterate.simps
tff(fact_7177_next_Osimps,axiom,
    ! [V3: code_natural,W2: code_natural] : aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),V3),W2)) = aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V3,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,V3,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2)))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W2,aa(num,code_natural,numeral_numeral(code_natural),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(aa(num,num,bit1,bit0(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,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,W2,aa(num,code_natural,numeral_numeral(code_natural),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))),one_one(code_natural)))),one_one(code_natural))),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,V3,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,V3,aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))))))))))))))),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),modulo_modulo(code_natural,W2,aa(num,code_natural,numeral_numeral(code_natural),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),bit0(bit0(aa(num,num,bit1,bit0(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,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),divide_divide(code_natural,W2,aa(num,code_natural,numeral_numeral(code_natural),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))) ).

% next.simps
tff(fact_7178_iterate_Opelims,axiom,
    ! [A: $tType,B: $tType,X: code_natural,Xa2: fun(B,fun(A,product_prod(B,A))),Xb: B,Y: fun(A,product_prod(B,A))] :
      ( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa2),Xb) = Y )
     => ( accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa2),Xb)))
       => ~ ( ( ( ( X = zero_zero(code_natural) )
               => ( Y = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb) ) )
              & ( ( X != zero_zero(code_natural) )
               => ( Y = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa2,Xb),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa2)) ) ) )
           => ~ accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa2),Xb))) ) ) ) ).

% iterate.pelims
tff(fact_7179_Random_Orange__def,axiom,
    ! [K2: code_natural] : range(K2) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),iterate(code_natural,product_prod(code_natural,code_natural),log(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),K2),aTP_Lamp_aii(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))))),one_one(code_natural)),aTP_Lamp_aij(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),K2)) ).

% Random.range_def
tff(fact_7180_Suc_Orep__eq,axiom,
    ! [X: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,code_Suc,X)) = aa(nat,nat,suc,aa(code_natural,nat,code_nat_of_natural,X)) ).

% Suc.rep_eq
tff(fact_7181_select__def,axiom,
    ! [A: $tType,Xs: list(A)] : select(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(aa(nat,code_natural,code_natural_of_nat,aa(list(A),nat,size_size(list(A)),Xs))),aTP_Lamp_aik(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).

% select_def
tff(fact_7182_trancl__def,axiom,
    ! [A: $tType,X5: set(product_prod(A,A))] : transitive_trancl(A,X5) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),transitive_tranclp(A,aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),X5)))) ).

% trancl_def
tff(fact_7183_Suc_Oabs__eq,axiom,
    ! [X: nat] : aa(code_natural,code_natural,code_Suc,aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,suc,X)) ).

% Suc.abs_eq
tff(fact_7184_less__eq__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] :
      ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa2)),aa(nat,code_natural,code_natural_of_nat,X)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa2),X)) ) ).

% less_eq_natural.abs_eq
tff(fact_7185_times__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa2)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xa2),X)) ).

% times_natural.abs_eq
tff(fact_7186_less__nat__rel,axiom,
    ord_less(nat) = transitive_tranclp(nat,aTP_Lamp_zu(nat,fun(nat,bool))) ).

% less_nat_rel
tff(fact_7187_zero__natural__def,axiom,
    zero_zero(code_natural) = aa(nat,code_natural,code_natural_of_nat,zero_zero(nat)) ).

% zero_natural_def
tff(fact_7188_tranclp__induct2,axiom,
    ! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Ax: A,Ay: B,Bx: A,By: B,P2: fun(A,fun(B,bool))] :
      ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_tranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
     => ( ! [A5: A,B4: B] :
            ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) )
       => ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
              ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_tranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)))
             => ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By)) ) ) ) ).

% tranclp_induct2
tff(fact_7189_tranclp__trancl__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X5: A,Xa: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),transitive_tranclp(A,aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X5),Xa))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa)),transitive_trancl(A,R))) ) ).

% tranclp_trancl_eq
tff(fact_7190_tranclp__power,axiom,
    ! [A: $tType,P2: fun(A,fun(A,bool)),X: A,Y: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),transitive_tranclp(A,P2),X),Y))
    <=> ? [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
          & pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N5),P2),X),Y)) ) ) ).

% tranclp_power
tff(fact_7191_Nitpick_Otranclp__unfold,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),transitive_tranclp(A,R),A3),B2))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).

% Nitpick.tranclp_unfold
tff(fact_7192_pick__same,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pick(A,map(A,product_prod(code_natural,A),aa(code_natural,fun(A,product_prod(code_natural,A)),product_Pair(code_natural,A),one_one(code_natural)),Xs),aa(nat,code_natural,code_natural_of_nat,L)) = aa(nat,A,nth(A,Xs),L) ) ) ).

% pick_same
tff(fact_7193_Code__Numeral_OSuc__def,axiom,
    code_Suc = aa(fun(nat,nat),fun(code_natural,code_natural),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat),suc) ).

% Code_Numeral.Suc_def
tff(fact_7194_times__natural__def,axiom,
    times_times(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),times_times(nat)) ).

% times_natural_def
tff(fact_7195_map__option__o__case__sum,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: fun(D,C),G3: fun(A,option(D)),H: fun(B,option(D))] : aa(fun(sum_sum(A,B),option(D)),fun(sum_sum(A,B),option(C)),comp(option(D),option(C),sum_sum(A,B),aa(fun(D,C),fun(option(D),option(C)),map_option(D,C),F3)),sum_case_sum(A,option(D),B,G3,H)) = sum_case_sum(A,option(C),B,aa(fun(A,option(D)),fun(A,option(C)),comp(option(D),option(C),A,aa(fun(D,C),fun(option(D),option(C)),map_option(D,C),F3)),G3),aa(fun(B,option(D)),fun(B,option(C)),comp(option(D),option(C),B,aa(fun(D,C),fun(option(D),option(C)),map_option(D,C),F3)),H)) ).

% map_option_o_case_sum
tff(fact_7196_semilattice__set_Oeq__fold_H,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),A6: set(A)] :
      ( lattic149705377957585745ce_set(A,F3)
     => ( lattic1715443433743089157tice_F(A,F3,A6) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_ail(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),F3),none(A),A6)) ) ) ).

% semilattice_set.eq_fold'
tff(fact_7197_linear__injective__on__subspace__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F3: fun(A,B),S2: set(A)] :
          ( real_Vector_linear(A,B,F3)
         => ( real_Vector_subspace(A,S2)
           => ( inj_on(A,B,F3,S2)
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
                 => ( ( aa(A,B,F3,X4) = zero_zero(B) )
                   => ( X4 = zero_zero(A) ) ) ) ) ) ) ) ).

% linear_injective_on_subspace_0
tff(fact_7198_subspace__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( real_Vector_subspace(A,S3)
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),S3)) ) ) ).

% subspace_0
tff(fact_7199_linear__subspace__kernel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_Vector_linear(A,B,F3)
         => real_Vector_subspace(A,aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aim(fun(A,B),fun(A,bool),F3))) ) ) ).

% linear_subspace_kernel
tff(fact_7200_subspace__single__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => real_Vector_subspace(A,aa(set(A),set(A),insert(A,zero_zero(A)),bot_bot(set(A)))) ) ).

% subspace_single_0
tff(fact_7201_subspaceI,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),S3))
         => ( ! [X3: A,Y3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y3),S3))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y3)),S3)) ) )
           => ( ! [C2: real,X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S3))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C2),X3)),S3)) )
             => real_Vector_subspace(A,S3) ) ) ) ) ).

% subspaceI
tff(fact_7202_subspace__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( real_Vector_subspace(A,S3)
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),S3))
            & ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
               => ! [Xa3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),S3))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Xa3)),S3)) ) )
            & ! [C4: real,X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S3))
               => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),C4),X4)),S3)) ) ) ) ) ).

% subspace_def
tff(fact_7203_old_Orec__bool__def,axiom,
    ! [T: $tType,X5: T,Xa: T,Xb3: bool] : product_rec_bool(T,X5,Xa,Xb3) = the(T,product_rec_set_bool(T,X5,Xa,Xb3)) ).

% old.rec_bool_def
tff(fact_7204_ordering__top_Oaxioms_I2_J,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ordering_top_axioms(A,Less_eq,Top) ) ).

% ordering_top.axioms(2)
tff(fact_7205_old_Obool_Osimps_I5_J,axiom,
    ! [T: $tType,F1: T,F22: T] : product_rec_bool(T,F1,F22,fTrue) = F1 ).

% old.bool.simps(5)
tff(fact_7206_old_Obool_Osimps_I6_J,axiom,
    ! [T: $tType,F1: T,F22: T] : product_rec_bool(T,F1,F22,fFalse) = F22 ).

% old.bool.simps(6)
tff(fact_7207_ordering__top__axioms__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Top: A] :
      ( ordering_top_axioms(A,Less_eq,Top)
    <=> ! [A7: A] : pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A7),Top)) ) ).

% ordering_top_axioms_def
tff(fact_7208_ordering__top__axioms_Ointro,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Top: A] :
      ( ! [A5: A] : pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),Top))
     => ordering_top_axioms(A,Less_eq,Top) ) ).

% ordering_top_axioms.intro
tff(fact_7209_ordering__top__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A] :
      ( ordering_top(A,Less_eq,Less,Top)
    <=> ( ordering(A,Less_eq,Less)
        & ordering_top_axioms(A,Less_eq,Top) ) ) ).

% ordering_top_def
tff(fact_7210_ordering__top_Ointro,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A] :
      ( ordering(A,Less_eq,Less)
     => ( ordering_top_axioms(A,Less_eq,Top)
       => ordering_top(A,Less_eq,Less,Top) ) ) ).

% ordering_top.intro
tff(fact_7211_ordering_Oeq__iff,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( ordering(A,Less_eq,Less)
     => ( ( A3 = B2 )
      <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
          & pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3)) ) ) ) ).

% ordering.eq_iff
tff(fact_7212_ordering_Oantisym,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( ordering(A,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3))
         => ( A3 = B2 ) ) ) ) ).

% ordering.antisym
tff(fact_7213_ordering_Oorder__iff__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( ordering(A,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
      <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
          | ( A3 = B2 ) ) ) ) ).

% ordering.order_iff_strict
tff(fact_7214_ordering_Ostrict__iff__order,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( ordering(A,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
      <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
          & ( A3 != B2 ) ) ) ) ).

% ordering.strict_iff_order
tff(fact_7215_ordering_Ostrict__implies__not__eq,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( ordering(A,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
       => ( A3 != B2 ) ) ) ).

% ordering.strict_implies_not_eq
tff(fact_7216_ordering_Onot__eq__order__implies__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( ordering(A,Less_eq,Less)
     => ( ( A3 != B2 )
       => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2)) ) ) ) ).

% ordering.not_eq_order_implies_strict
tff(fact_7217_ordering__strictI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( ! [A5: A,B4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
            | ( A5 = B4 ) ) )
     => ( ! [A5: A,B4: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,B4),A5)) )
       => ( ! [A5: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),A5))
         => ( ! [A5: A,B4: A,C2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B4),C2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),C2)) ) )
           => ordering(A,Less_eq,Less) ) ) ) ) ).

% ordering_strictI
tff(fact_7218_ordering__dualI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( ordering(A,aTP_Lamp_ain(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less_eq),aTP_Lamp_ain(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less))
     => ordering(A,Less_eq,Less) ) ).

% ordering_dualI
tff(fact_7219_ordering__top_Oaxioms_I1_J,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ordering(A,Less_eq,Less) ) ).

% ordering_top.axioms(1)
tff(fact_7220_order_Oordering__axioms,axiom,
    ! [A: $tType] :
      ( order(A)
     => ordering(A,ord_less_eq(A),ord_less(A)) ) ).

% order.ordering_axioms
tff(fact_7221_dual__order_Oordering__axioms,axiom,
    ! [A: $tType] :
      ( order(A)
     => ordering(A,aTP_Lamp_aio(A,fun(A,bool)),aTP_Lamp_aip(A,fun(A,bool))) ) ).

% dual_order.ordering_axioms
tff(fact_7222_eventually__filtercomap__at__bot__linorder,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P2: fun(B,bool),F3: fun(B,A)] :
          ( eventually(B,P2,filtercomap(B,A,F3,at_bot(A)))
        <=> ? [N7: A] :
            ! [X4: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X4)),N7))
             => pp(aa(B,bool,P2,X4)) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_7223_list__all__length,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] :
      ( list_all(A,P2,Xs)
    <=> ! [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P2,aa(nat,A,nth(A,Xs),N5))) ) ) ).

% list_all_length
tff(fact_7224_eventually__filtercomap__at__top__linorder,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P2: fun(B,bool),F3: fun(B,A)] :
          ( eventually(B,P2,filtercomap(B,A,F3,at_top(A)))
        <=> ? [N7: A] :
            ! [X4: B] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N7),aa(B,A,F3,X4)))
             => pp(aa(B,bool,P2,X4)) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_7225_prod__list_Ocomm__monoid__list__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => groups1828464146339083142d_list(A,times_times(A),one_one(A)) ) ).

% prod_list.comm_monoid_list_axioms
tff(fact_7226_Pair__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A6: fun(A,fun(B,bool)),B6: fun(C,fun(D,bool))] : pp(aa(fun(B,fun(D,product_prod(B,D))),bool,aa(fun(A,fun(C,product_prod(A,C))),fun(fun(B,fun(D,product_prod(B,D))),bool),bNF_rel_fun(A,B,fun(C,product_prod(A,C)),fun(D,product_prod(B,D)),A6,bNF_rel_fun(C,D,product_prod(A,C),product_prod(B,D),B6,basic_rel_prod(A,B,C,D,A6,B6))),product_Pair(A,C)),product_Pair(B,D))) ).

% Pair_transfer
tff(fact_7227_sum__list_Ocomm__monoid__list__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => groups1828464146339083142d_list(A,plus_plus(A),zero_zero(A)) ) ).

% sum_list.comm_monoid_list_axioms
tff(fact_7228_VEBT_Orec__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,bool))] : pp(aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B))))),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B))),bool,aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A))),fun(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B))))),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B))),bool),bNF_rel_fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B)),bNF_rel_fun(option(product_prod(nat,nat)),option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A)))),fun(nat,fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B)))),fequal(option(product_prod(nat,nat))),bNF_rel_fun(nat,nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))),fun(list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(B,B))),fequal(nat),bNF_rel_fun(list(product_prod(vEBT_VEBT,A)),list(product_prod(vEBT_VEBT,B)),fun(vEBT_VEBT,fun(A,A)),fun(vEBT_VEBT,fun(B,B)),list_all2(product_prod(vEBT_VEBT,A),product_prod(vEBT_VEBT,B),basic_rel_prod(vEBT_VEBT,vEBT_VEBT,A,B,fequal(vEBT_VEBT),S3)),bNF_rel_fun(vEBT_VEBT,vEBT_VEBT,fun(A,A),fun(B,B),fequal(vEBT_VEBT),bNF_rel_fun(A,B,A,B,S3,S3))))),bNF_rel_fun(fun(bool,fun(bool,A)),fun(bool,fun(bool,B)),fun(vEBT_VEBT,A),fun(vEBT_VEBT,B),bNF_rel_fun(bool,bool,fun(bool,A),fun(bool,B),fequal(bool),bNF_rel_fun(bool,bool,A,B,fequal(bool),S3)),bNF_rel_fun(vEBT_VEBT,vEBT_VEBT,A,B,fequal(vEBT_VEBT),S3))),vEBT_rec_VEBT(A)),vEBT_rec_VEBT(B))) ).

% VEBT.rec_transfer
tff(fact_7229_rel__prod__inject,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,bool)),R23: fun(C,fun(D,bool)),A3: A,B2: C,C3: B,D3: D] :
      ( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R12,R23),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B2)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),C3),D3)))
    <=> ( pp(aa(B,bool,aa(A,fun(B,bool),R12,A3),C3))
        & pp(aa(D,bool,aa(C,fun(D,bool),R23,B2),D3)) ) ) ).

% rel_prod_inject
tff(fact_7230_rel__prod_Ocases,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,bool)),R23: fun(C,fun(D,bool)),A12: product_prod(A,C),A23: product_prod(B,D)] :
      ( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R12,R23),A12),A23))
     => ~ ! [A5: A,B4: B,C2: C] :
            ( ( A12 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A5),C2) )
           => ! [D2: D] :
                ( ( A23 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B4),D2) )
               => ( pp(aa(B,bool,aa(A,fun(B,bool),R12,A5),B4))
                 => ~ pp(aa(D,bool,aa(C,fun(D,bool),R23,C2),D2)) ) ) ) ) ).

% rel_prod.cases
tff(fact_7231_rel__prod_Osimps,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,bool)),R23: fun(C,fun(D,bool)),A12: product_prod(A,C),A23: product_prod(B,D)] :
      ( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R12,R23),A12),A23))
    <=> ? [A7: A,B5: B,C4: C,D4: D] :
          ( ( A12 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A7),C4) )
          & ( A23 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B5),D4) )
          & pp(aa(B,bool,aa(A,fun(B,bool),R12,A7),B5))
          & pp(aa(D,bool,aa(C,fun(D,bool),R23,C4),D4)) ) ) ).

% rel_prod.simps
tff(fact_7232_rel__prod_Ointros,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,R12: fun(A,fun(B,bool)),A3: A,B2: B,R23: fun(C,fun(D,bool)),C3: C,D3: D] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R12,A3),B2))
     => ( pp(aa(D,bool,aa(C,fun(D,bool),R23,C3),D3))
       => pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R12,R23),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C3)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B2),D3))) ) ) ).

% rel_prod.intros
tff(fact_7233_VEBT_Ocase__transfer,axiom,
    ! [A: $tType,B: $tType,S3: fun(A,fun(B,bool))] : pp(aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B))),bool,aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A))),fun(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B))),bool),bNF_rel_fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B)),bNF_rel_fun(option(product_prod(nat,nat)),option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B))),fequal(option(product_prod(nat,nat))),bNF_rel_fun(nat,nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)),fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)),fequal(nat),bNF_rel_fun(list(vEBT_VEBT),list(vEBT_VEBT),fun(vEBT_VEBT,A),fun(vEBT_VEBT,B),fequal(list(vEBT_VEBT)),bNF_rel_fun(vEBT_VEBT,vEBT_VEBT,A,B,fequal(vEBT_VEBT),S3)))),bNF_rel_fun(fun(bool,fun(bool,A)),fun(bool,fun(bool,B)),fun(vEBT_VEBT,A),fun(vEBT_VEBT,B),bNF_rel_fun(bool,bool,fun(bool,A),fun(bool,B),fequal(bool),bNF_rel_fun(bool,bool,A,B,fequal(bool),S3)),bNF_rel_fun(vEBT_VEBT,vEBT_VEBT,A,B,fequal(vEBT_VEBT),S3))),vEBT_case_VEBT(A)),vEBT_case_VEBT(B))) ).

% VEBT.case_transfer
tff(fact_7234_is__arg__max__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),P2: fun(A,bool),X: A] :
          ( lattic501386751176901750rg_max(A,B,F3,P2,X)
        <=> ( pp(aa(A,bool,P2,X))
            & ! [Y5: A] :
                ( pp(aa(A,bool,P2,Y5))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y5)),aa(A,B,F3,X))) ) ) ) ) ).

% is_arg_max_linorder
tff(fact_7235_VEBT_Osimps_I6_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F22: fun(bool,fun(bool,A)),X21: bool,X22: bool] : aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_case_VEBT(A),F1),F22),vEBT_Leaf(X21,X22)) = aa(bool,A,aa(bool,fun(bool,A),F22,X21),X22) ).

% VEBT.simps(6)
tff(fact_7236_VEBT_Osimps_I5_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F22: fun(bool,fun(bool,A)),X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_case_VEBT(A),F1),F22),vEBT_Node(X11,X12,X13,X14)) = aa(vEBT_VEBT,A,aa(list(vEBT_VEBT),fun(vEBT_VEBT,A),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))),F1,X11),X12),X13),X14) ).

% VEBT.simps(5)
tff(fact_7237_VEBT_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F22: fun(bool,fun(bool,A)),VEBT: vEBT_VEBT] : aa(A,B,H,aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_case_VEBT(A),F1),F22),VEBT)) = aa(vEBT_VEBT,B,aa(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(fun(bool,fun(bool,B)),fun(vEBT_VEBT,B)),vEBT_case_VEBT(B),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),aTP_Lamp_aiq(fun(A,B),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B))))),H),F1)),aa(fun(bool,fun(bool,A)),fun(bool,fun(bool,B)),aTP_Lamp_air(fun(A,B),fun(fun(bool,fun(bool,A)),fun(bool,fun(bool,B))),H),F22)),VEBT) ).

% VEBT.case_distrib
tff(fact_7238_group_Oaxioms_I2_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group(A,F3,Z2,Inverse)
     => group_axioms(A,F3,Z2,Inverse) ) ).

% group.axioms(2)
tff(fact_7239_is__none__bind,axiom,
    ! [A: $tType,B: $tType,F3: option(B),G3: fun(B,option(A))] :
      ( is_none(A,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),F3),G3))
    <=> ( is_none(B,F3)
        | is_none(A,aa(B,option(A),G3,aa(option(B),B,the2(B),F3))) ) ) ).

% is_none_bind
tff(fact_7240_is__none__code_I2_J,axiom,
    ! [B: $tType,X: B] : ~ is_none(B,aa(B,option(B),some(B),X)) ).

% is_none_code(2)
tff(fact_7241_is__none__code_I1_J,axiom,
    ! [A: $tType] : is_none(A,none(A)) ).

% is_none_code(1)
tff(fact_7242_is__none__map__option,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X: option(B)] :
      ( is_none(A,aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F3),X))
    <=> is_none(B,X) ) ).

% is_none_map_option
tff(fact_7243_group__axioms__def,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group_axioms(A,F3,Z2,Inverse)
    <=> ( ! [A7: A] : aa(A,A,aa(A,fun(A,A),F3,Z2),A7) = A7
        & ! [A7: A] : aa(A,A,aa(A,fun(A,A),F3,aa(A,A,Inverse,A7)),A7) = Z2 ) ) ).

% group_axioms_def
tff(fact_7244_group__axioms_Ointro,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F3,Z2),A5) = A5
     => ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F3,aa(A,A,Inverse,A5)),A5) = Z2
       => group_axioms(A,F3,Z2,Inverse) ) ) ).

% group_axioms.intro
tff(fact_7245_Option_Ois__none__def,axiom,
    ! [A: $tType,X: option(A)] :
      ( is_none(A,X)
    <=> ( X = none(A) ) ) ).

% Option.is_none_def
tff(fact_7246_is__none__simps_I1_J,axiom,
    ! [A: $tType] : is_none(A,none(A)) ).

% is_none_simps(1)
tff(fact_7247_is__none__simps_I2_J,axiom,
    ! [B: $tType,X: B] : ~ is_none(B,aa(B,option(B),some(B),X)) ).

% is_none_simps(2)
tff(fact_7248_the__map__option,axiom,
    ! [B: $tType,A: $tType,X: option(A),F3: fun(A,B)] :
      ( ~ is_none(A,X)
     => ( aa(option(B),B,the2(B),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F3),X)) = aa(A,B,F3,aa(option(A),A,the2(A),X)) ) ) ).

% the_map_option
tff(fact_7249_rel__option__unfold,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X: option(A),Y: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R2),X),Y))
    <=> ( ( is_none(A,X)
        <=> is_none(B,Y) )
        & ( ~ is_none(A,X)
         => ( ~ is_none(B,Y)
           => pp(aa(B,bool,aa(A,fun(B,bool),R2,aa(option(A),A,the2(A),X)),aa(option(B),B,the2(B),Y))) ) ) ) ) ).

% rel_option_unfold
tff(fact_7250_rel__optionI,axiom,
    ! [A: $tType,B: $tType,X: option(A),Y: option(B),P2: fun(A,fun(B,bool))] :
      ( ( is_none(A,X)
      <=> is_none(B,Y) )
     => ( ( ~ is_none(A,X)
         => ( ~ is_none(B,Y)
           => pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(option(A),A,the2(A),X)),aa(option(B),B,the2(B),Y))) ) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),P2),X),Y)) ) ) ).

% rel_optionI
tff(fact_7251_group_Ointro,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( semigroup(A,F3)
     => ( group_axioms(A,F3,Z2,Inverse)
       => group(A,F3,Z2,Inverse) ) ) ).

% group.intro
tff(fact_7252_group__def,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group(A,F3,Z2,Inverse)
    <=> ( semigroup(A,F3)
        & group_axioms(A,F3,Z2,Inverse) ) ) ).

% group_def
tff(fact_7253_add_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => semigroup(A,plus_plus(A)) ) ).

% add.semigroup_axioms
tff(fact_7254_semigroup_Oassoc,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),A3: A,B2: A,C3: A] :
      ( semigroup(A,F3)
     => ( aa(A,A,aa(A,fun(A,A),F3,aa(A,A,aa(A,fun(A,A),F3,A3),B2)),C3) = aa(A,A,aa(A,fun(A,A),F3,A3),aa(A,A,aa(A,fun(A,A),F3,B2),C3)) ) ) ).

% semigroup.assoc
tff(fact_7255_semigroup_Ointro,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A))] :
      ( ! [A5: A,B4: A,C2: A] : aa(A,A,aa(A,fun(A,A),F3,aa(A,A,aa(A,fun(A,A),F3,A5),B4)),C2) = aa(A,A,aa(A,fun(A,A),F3,A5),aa(A,A,aa(A,fun(A,A),F3,B4),C2))
     => semigroup(A,F3) ) ).

% semigroup.intro
tff(fact_7256_semigroup__def,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A))] :
      ( semigroup(A,F3)
    <=> ! [A7: A,B5: A,C4: A] : aa(A,A,aa(A,fun(A,A),F3,aa(A,A,aa(A,fun(A,A),F3,A7),B5)),C4) = aa(A,A,aa(A,fun(A,A),F3,A7),aa(A,A,aa(A,fun(A,A),F3,B5),C4)) ) ).

% semigroup_def
tff(fact_7257_group_Oaxioms_I1_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
      ( group(A,F3,Z2,Inverse)
     => semigroup(A,F3) ) ).

% group.axioms(1)
tff(fact_7258_mult_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => semigroup(A,times_times(A)) ) ).

% mult.semigroup_axioms
tff(fact_7259_ordering_Oaxioms_I2_J,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( ordering(A,Less_eq,Less)
     => ordering_axioms(A,Less_eq,Less) ) ).

% ordering.axioms(2)
tff(fact_7260_case__prod__curry,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),product_curry(A,B,C,F3)) = F3 ).

% case_prod_curry
tff(fact_7261_curryI,axiom,
    ! [A: $tType,B: $tType,F3: fun(product_prod(A,B),bool),A3: A,B2: B] :
      ( pp(aa(product_prod(A,B),bool,F3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
     => pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F3),A3),B2)) ) ).

% curryI
tff(fact_7262_curry__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(product_prod(B,C),A),A3: B,B2: C] : aa(C,A,aa(B,fun(C,A),product_curry(B,C,A,F3),A3),B2) = aa(product_prod(B,C),A,F3,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) ).

% curry_conv
tff(fact_7263_curry__case__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C))] : product_curry(A,B,C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3)) = F3 ).

% curry_case_prod
tff(fact_7264_curry__def,axiom,
    ! [C: $tType,A: $tType,B: $tType,X5: fun(product_prod(A,B),C),Xa: A,Xb3: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,X5),Xa),Xb3) = aa(product_prod(A,B),C,X5,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb3)) ).

% curry_def
tff(fact_7265_curryE,axiom,
    ! [A: $tType,B: $tType,F3: fun(product_prod(A,B),bool),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F3),A3),B2))
     => pp(aa(product_prod(A,B),bool,F3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).

% curryE
tff(fact_7266_curryD,axiom,
    ! [A: $tType,B: $tType,F3: fun(product_prod(A,B),bool),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F3),A3),B2))
     => pp(aa(product_prod(A,B),bool,F3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).

% curryD
tff(fact_7267_curry__K,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: C,X5: A,Xa: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,aTP_Lamp_ais(C,fun(product_prod(A,B),C),C3)),X5),Xa) = C3 ).

% curry_K
tff(fact_7268_ordering__axioms__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( ordering_axioms(A,Less_eq,Less)
    <=> ( ! [A7: A,B5: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A7),B5))
          <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A7),B5))
              & ( A7 != B5 ) ) )
        & ! [A7: A,B5: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A7),B5))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B5),A7))
             => ( A7 = B5 ) ) ) ) ) ).

% ordering_axioms_def
tff(fact_7269_ordering__axioms_Ointro,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),Less_eq: fun(A,fun(A,bool))] :
      ( ! [A5: A,B4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
            & ( A5 != B4 ) ) )
     => ( ! [A5: A,B4: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B4),A5))
             => ( A5 = B4 ) ) )
       => ordering_axioms(A,Less_eq,Less) ) ) ).

% ordering_axioms.intro
tff(fact_7270_ordering_Ointro,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( partial_preordering(A,Less_eq)
     => ( ordering_axioms(A,Less_eq,Less)
       => ordering(A,Less_eq,Less) ) ) ).

% ordering.intro
tff(fact_7271_ordering__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( ordering(A,Less_eq,Less)
    <=> ( partial_preordering(A,Less_eq)
        & ordering_axioms(A,Less_eq,Less) ) ) ).

% ordering_def
tff(fact_7272_partial__preordering_Orefl,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A] :
      ( partial_preordering(A,Less_eq)
     => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),A3)) ) ).

% partial_preordering.refl
tff(fact_7273_partial__preordering_Ointro,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] :
      ( ! [A5: A] : pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),A5))
     => ( ! [A5: A,B4: A,C2: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B4),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),C2)) ) )
       => partial_preordering(A,Less_eq) ) ) ).

% partial_preordering.intro
tff(fact_7274_partial__preordering_Otrans,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A,B2: A,C3: A] :
      ( partial_preordering(A,Less_eq)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C3)) ) ) ) ).

% partial_preordering.trans
tff(fact_7275_partial__preordering__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] :
      ( partial_preordering(A,Less_eq)
    <=> ( ! [A7: A] : pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A7),A7))
        & ! [A7: A,B5: A,C4: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A7),B5))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B5),C4))
             => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A7),C4)) ) ) ) ) ).

% partial_preordering_def
tff(fact_7276_dual__order_Opartial__preordering__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => partial_preordering(A,aTP_Lamp_aeh(A,fun(A,bool))) ) ).

% dual_order.partial_preordering_axioms
tff(fact_7277_order_Opartial__preordering__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => partial_preordering(A,ord_less_eq(A)) ) ).

% order.partial_preordering_axioms
tff(fact_7278_ordering_Oaxioms_I1_J,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( ordering(A,Less_eq,Less)
     => partial_preordering(A,Less_eq) ) ).

% ordering.axioms(1)
tff(fact_7279_wfP__wf__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wfP(A,aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
    <=> wf(A,R) ) ).

% wfP_wf_eq
tff(fact_7280_prod__list_Omonoid__list__axioms,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => groups_monoid_list(A,times_times(A),one_one(A)) ) ).

% prod_list.monoid_list_axioms
tff(fact_7281_sum__list_Omonoid__list__axioms,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => groups_monoid_list(A,plus_plus(A),zero_zero(A)) ) ).

% sum_list.monoid_list_axioms
tff(fact_7282_typerep_Osize__neq,axiom,
    ! [X: typerep] : aa(typerep,nat,size_size(typerep),X) != zero_zero(nat) ).

% typerep.size_neq
tff(fact_7283_tuple__isomorphism_Osize__neq,axiom,
    ! [A: $tType,B: $tType,C: $tType,X: tuple_isomorphism(A,B,C)] : aa(tuple_isomorphism(A,B,C),nat,size_size(tuple_isomorphism(A,B,C)),X) != zero_zero(nat) ).

% tuple_isomorphism.size_neq
tff(fact_7284_tuple__isomorphism_Osize_I2_J,axiom,
    ! [B: $tType,C: $tType,A: $tType,X1: fun(A,product_prod(B,C)),X2: fun(product_prod(B,C),A)] : aa(tuple_isomorphism(A,B,C),nat,size_size(tuple_isomorphism(A,B,C)),tuple_1188178415141063261rphism(A,B,C,X1,X2)) = aa(nat,nat,suc,zero_zero(nat)) ).

% tuple_isomorphism.size(2)
tff(fact_7285_typerep_Osize_I2_J,axiom,
    ! [X1: literal,X2: list(typerep)] : aa(typerep,nat,size_size(typerep),typerep2(X1,X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(typerep,size_size(typerep),X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% typerep.size(2)
tff(fact_7286_typerep_Osize__gen,axiom,
    ! [X1: literal,X2: list(typerep)] : aa(typerep,nat,size_typerep,typerep2(X1,X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(typerep,size_typerep,X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% typerep.size_gen
tff(fact_7287_tuple__isomorphism_Osize__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType,Xb: fun(A,nat),Xa2: fun(B,nat),X: fun(C,nat),X1: fun(A,product_prod(B,C)),X2: fun(product_prod(B,C),A)] : tuple_9181185373184732606rphism(A,B,C,Xb,Xa2,X,tuple_1188178415141063261rphism(A,B,C,X1,X2)) = aa(nat,nat,suc,zero_zero(nat)) ).

% tuple_isomorphism.size_gen
tff(fact_7288_max__nat_Osemilattice__neutr__axioms,axiom,
    semilattice_neutr(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.semilattice_neutr_axioms
tff(fact_7289_curr__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,A6: set(A),F3: fun(product_prod(A,B),C),X5: A] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A6))
       => ( bNF_Wellorder_curr(A,B,C,A6,F3,X5) = aa(A,fun(B,C),aTP_Lamp_hw(fun(product_prod(A,B),C),fun(A,fun(B,C)),F3),X5) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X5),A6))
       => ( bNF_Wellorder_curr(A,B,C,A6,F3,X5) = undefined(fun(B,C)) ) ) ) ).

% curr_def
tff(fact_7290_gcd__nat_Osemilattice__neutr__axioms,axiom,
    semilattice_neutr(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.semilattice_neutr_axioms
tff(fact_7291_or_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => semilattice_neutr(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.semilattice_neutr_axioms
tff(fact_7292_dual__order_Opreordering__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => preordering(A,aTP_Lamp_aeh(A,fun(A,bool)),aTP_Lamp_aig(A,fun(A,bool))) ) ).

% dual_order.preordering_axioms
tff(fact_7293_closed__Collect__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F3)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G3)
           => topolo7761053866217962861closed(A,aa(fun(A,bool),set(A),collect(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_ait(fun(A,B),fun(fun(A,B),fun(A,bool)),F3),G3))) ) ) ) ).

% closed_Collect_le
tff(fact_7294_preordering_Oasym,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( preordering(A,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
       => ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),A3)) ) ) ).

% preordering.asym
tff(fact_7295_preordering_Oirrefl,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A] :
      ( preordering(A,Less_eq,Less)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),A3)) ) ).

% preordering.irrefl
tff(fact_7296_preordering_Ostrict__trans,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C3: A] :
      ( preordering(A,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
       => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C3)) ) ) ) ).

% preordering.strict_trans
tff(fact_7297_preordering_Ostrict__trans1,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C3: A] :
      ( preordering(A,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
       => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C3)) ) ) ) ).

% preordering.strict_trans1
tff(fact_7298_preordering_Ostrict__trans2,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C3: A] :
      ( preordering(A,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
       => ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),C3))
         => pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C3)) ) ) ) ).

% preordering.strict_trans2
tff(fact_7299_preordering_Ostrict__iff__not,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( preordering(A,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
      <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
          & ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3)) ) ) ) ).

% preordering.strict_iff_not
tff(fact_7300_preordering_Ostrict__implies__order,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( preordering(A,Less_eq,Less)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
       => pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2)) ) ) ).

% preordering.strict_implies_order
tff(fact_7301_preordering__strictI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( ! [A5: A,B4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
            | ( A5 = B4 ) ) )
     => ( ! [A5: A,B4: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,B4),A5)) )
       => ( ! [A5: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),A5))
         => ( ! [A5: A,B4: A,C2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B4),C2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),C2)) ) )
           => preordering(A,Less_eq,Less) ) ) ) ) ).

% preordering_strictI
tff(fact_7302_preordering__dualI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( preordering(A,aTP_Lamp_ain(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less_eq),aTP_Lamp_ain(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less))
     => preordering(A,Less_eq,Less) ) ).

% preordering_dualI
tff(fact_7303_order_Opreordering__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => preordering(A,ord_less_eq(A),ord_less(A)) ) ).

% order.preordering_axioms
tff(fact_7304_closed__diagonal,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_aiu(product_prod(A,A),bool))) ) ).

% closed_diagonal
tff(fact_7305_preordering_Oaxioms_I1_J,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( preordering(A,Less_eq,Less)
     => partial_preordering(A,Less_eq) ) ).

% preordering.axioms(1)
tff(fact_7306_closed__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_aiv(product_prod(A,A),bool))) ) ).

% closed_subdiagonal
tff(fact_7307_closed__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_aiw(product_prod(A,A),bool))) ) ).

% closed_superdiagonal
tff(fact_7308_preordering__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( preordering(A,Less_eq,Less)
    <=> ( partial_preordering(A,Less_eq)
        & preordering_axioms(A,Less_eq,Less) ) ) ).

% preordering_def
tff(fact_7309_preordering_Ointro,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( partial_preordering(A,Less_eq)
     => ( preordering_axioms(A,Less_eq,Less)
       => preordering(A,Less_eq,Less) ) ) ).

% preordering.intro
tff(fact_7310_preordering__axioms__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( preordering_axioms(A,Less_eq,Less)
    <=> ! [A7: A,B5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A7),B5))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A7),B5))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B5),A7)) ) ) ) ).

% preordering_axioms_def
tff(fact_7311_preordering__axioms_Ointro,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool)),Less_eq: fun(A,fun(A,bool))] :
      ( ! [A5: A,B4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A5),B4))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A5),B4))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B4),A5)) ) )
     => preordering_axioms(A,Less_eq,Less) ) ).

% preordering_axioms.intro
tff(fact_7312_preordering_Oaxioms_I2_J,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( preordering(A,Less_eq,Less)
     => preordering_axioms(A,Less_eq,Less) ) ).

% preordering.axioms(2)
tff(fact_7313_real__times__code,axiom,
    ! [X: rat,Y: rat] : aa(real,real,aa(real,fun(real,real),times_times(real),ratreal(X)),ratreal(Y)) = ratreal(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),X),Y)) ).

% real_times_code
tff(fact_7314_single__valued__confluent,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
      ( single_valued(A,A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,R)))
       => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),transitive_rtrancl(A,R)))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_rtrancl(A,R)))
            | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Y)),transitive_rtrancl(A,R))) ) ) ) ) ).

% single_valued_confluent
tff(fact_7315_single__valued__def,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,B))] :
      ( single_valued(A,B,R)
    <=> ! [X4: A,Y5: B] :
          ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y5)),R))
         => ! [Z3: B] :
              ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z3)),R))
             => ( Y5 = Z3 ) ) ) ) ).

% single_valued_def
tff(fact_7316_single__valuedI,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
      ( ! [X3: A,Y3: B,Z: B] :
          ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y3)),R))
         => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z)),R))
           => ( Y3 = Z ) ) )
     => single_valued(A,B,R) ) ).

% single_valuedI
tff(fact_7317_single__valuedD,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,B)),X: A,Y: B,Z2: B] :
      ( single_valued(A,B,R)
     => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),R))
       => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z2)),R))
         => ( Y = Z2 ) ) ) ) ).

% single_valuedD
tff(fact_7318_single__valuedp__single__valued__eq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
      ( single_valuedp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),R))
    <=> single_valued(A,B,R) ) ).

% single_valuedp_single_valued_eq
tff(fact_7319_comm__monoid__set_Ozero__middle,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,P: nat,K2: nat,G3: fun(nat,A),H: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),P))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aa(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),aTP_Lamp_aix(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Z2),K2),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),P)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_aiy(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% comm_monoid_set.zero_middle
tff(fact_7320_comm__monoid__set_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(B)
     => ! [F3: fun(A,fun(A,A)),Z2: A,A3: B,C3: B,B2: B,D3: B,G3: fun(B,A),H: fun(B,A)] :
          ( groups778175481326437816id_set(A,F3,Z2)
         => ( ( A3 = C3 )
           => ( ( B2 = D3 )
             => ( ! [X3: B] :
                    ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C3),X3))
                   => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D3))
                     => ( aa(B,A,G3,X3) = aa(B,A,H,X3) ) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F3,Z2),G3),set_or7035219750837199246ssThan(B,A3,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F3,Z2),H),set_or7035219750837199246ssThan(B,C3,D3)) ) ) ) ) ) ) ).

% comm_monoid_set.ivl_cong
tff(fact_7321_comm__monoid__set_Ohead,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,M2: nat,N: nat,G3: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),F3,aa(nat,A,G3,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or3652927894154168847AtMost(nat,M2,N))) ) ) ) ).

% comm_monoid_set.head
tff(fact_7322_comm__monoid__set_Olast__plus,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,M2: nat,N: nat,G3: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),F3,aa(nat,A,G3,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,M2,N))) ) ) ) ).

% comm_monoid_set.last_plus
tff(fact_7323_comm__monoid__set_Onat__ivl__Suc_H,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,M2: nat,N: nat,G3: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,suc,N)))
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F3,aa(nat,A,G3,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).

% comm_monoid_set.nat_ivl_Suc'
tff(fact_7324_comm__monoid__set_OatLeast__Suc__atMost,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,M2: nat,N: nat,G3: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),F3,aa(nat,A,G3,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% comm_monoid_set.atLeast_Suc_atMost
tff(fact_7325_comm__monoid__set_OSuc__reindex__ivl,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,M2: nat,N: nat,G3: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( aa(A,A,aa(A,fun(A,A),F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,N))),aa(nat,A,G3,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F3,aa(nat,A,G3,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aTP_Lamp_mh(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M2,N))) ) ) ) ).

% comm_monoid_set.Suc_reindex_ivl
tff(fact_7326_comm__monoid__set_OatLeastLessThan__concat,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,M2: nat,N: nat,P: nat,G3: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P))
         => ( aa(A,A,aa(A,fun(A,A),F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,M2,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,N,P))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,M2,P)) ) ) ) ) ).

% comm_monoid_set.atLeastLessThan_concat
tff(fact_7327_comm__monoid__set_OatLeastLessThan__Suc,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,A3: nat,B2: nat,G3: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% comm_monoid_set.atLeastLessThan_Suc
tff(fact_7328_comm__monoid__set_Onat__diff__reindex,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(nat,fun(nat,A),aTP_Lamp_aiz(fun(nat,A),fun(nat,fun(nat,A)),G3),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).

% comm_monoid_set.nat_diff_reindex
tff(fact_7329_prod_Ocomm__monoid__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => groups778175481326437816id_set(A,times_times(A),one_one(A)) ) ).

% prod.comm_monoid_set_axioms
tff(fact_7330_comm__monoid__mult__class_Oprod__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ( groups7121269368397514597t_prod(B,A) = groups_comm_monoid_F(A,B,times_times(A),one_one(A)) ) ) ).

% comm_monoid_mult_class.prod_def
tff(fact_7331_comm__monoid__set_Ohead__if,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,N: nat,M2: nat,G3: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,N)) = Z2 ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G3,N)) ) ) ) ) ).

% comm_monoid_set.head_if
tff(fact_7332_comm__monoid__set_Ocl__ivl__Suc,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,N: nat,M2: nat,G3: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = Z2 ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ) ) ) ).

% comm_monoid_set.cl_ivl_Suc
tff(fact_7333_comm__monoid__set_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,M2: nat,N: nat,G3: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,M2,N)) = aa(A,A,aa(A,fun(A,A),F3,aa(nat,A,G3,M2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),N))) ) ) ) ).

% comm_monoid_set.atLeast_Suc_lessThan
tff(fact_7334_comm__monoid__set_Oop__ivl__Suc,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,N: nat,M2: nat,G3: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = Z2 ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,M2,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,M2,N))),aa(nat,A,G3,N)) ) ) ) ) ).

% comm_monoid_set.op_ivl_Suc
tff(fact_7335_comm__monoid__set_OatMost__Suc,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ) ).

% comm_monoid_set.atMost_Suc
tff(fact_7336_comm__monoid__set_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aTP_Lamp_mh(fun(nat,A),fun(nat,A),G3)),set_or7035219750837199246ssThan(nat,M2,N)) ) ) ).

% comm_monoid_set.shift_bounds_Suc_ivl
tff(fact_7337_comm__monoid__set_OlessThan__Suc,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,A,G3,N)) ) ) ).

% comm_monoid_set.lessThan_Suc
tff(fact_7338_comm__monoid__set_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aTP_Lamp_mh(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ).

% comm_monoid_set.shift_bounds_cl_Suc_ivl
tff(fact_7339_comm__monoid__set_Onested__swap_H,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,A3: fun(nat,fun(nat,A)),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_aja(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F3),Z2),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(nat,fun(nat,A),aa(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),aTP_Lamp_ajc(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),F3),Z2),A3),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).

% comm_monoid_set.nested_swap'
tff(fact_7340_comm__monoid__set_Onested__swap,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,A3: fun(nat,fun(nat,A)),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_ajd(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F3),Z2),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(nat,fun(nat,A),aa(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),aTP_Lamp_ajc(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),F3),Z2),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).

% comm_monoid_set.nested_swap
tff(fact_7341_comm__monoid__set_OatLeast1__atMost__eq,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aTP_Lamp_mh(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).

% comm_monoid_set.atLeast1_atMost_eq
tff(fact_7342_comm__monoid__set_OatMost__shift,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),F3,aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aTP_Lamp_mh(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).

% comm_monoid_set.atMost_shift
tff(fact_7343_comm__monoid__set_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G3,aa(nat,nat,suc,N))) ) ) ).

% comm_monoid_set.atLeast0_atMost_Suc
tff(fact_7344_comm__monoid__set_OlessThan__Suc__shift,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F3,aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aTP_Lamp_mh(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).

% comm_monoid_set.lessThan_Suc_shift
tff(fact_7345_comm__monoid__set_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G3,N)) ) ) ).

% comm_monoid_set.atLeast0_lessThan_Suc
tff(fact_7346_comm__monoid__set_OatMost__Suc__shift,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F3,aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aTP_Lamp_mh(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ) ).

% comm_monoid_set.atMost_Suc_shift
tff(fact_7347_sum_Ocomm__monoid__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => groups778175481326437816id_set(A,plus_plus(A),zero_zero(A)) ) ).

% sum.comm_monoid_set_axioms
tff(fact_7348_comm__monoid__add__class_Osum__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ( groups7311177749621191930dd_sum(B,A) = groups_comm_monoid_F(A,B,plus_plus(A),zero_zero(A)) ) ) ).

% comm_monoid_add_class.sum_def
tff(fact_7349_comm__monoid__set_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),M2: nat,K2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(nat,fun(nat,A),aTP_Lamp_aje(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or7035219750837199246ssThan(nat,M2,N)) ) ) ).

% comm_monoid_set.shift_bounds_nat_ivl
tff(fact_7350_comm__monoid__set_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),M2: nat,K2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(nat,fun(nat,A),aTP_Lamp_aje(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ).

% comm_monoid_set.shift_bounds_cl_nat_ivl
tff(fact_7351_comm__monoid__set_Oin__pairs__0,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,A),fun(nat,A),aTP_Lamp_ajf(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),F3),G3)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% comm_monoid_set.in_pairs_0
tff(fact_7352_comm__monoid__set_OatLeastLessThan__rev,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat,M2: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,N,M2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ajg(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M2)),set_or7035219750837199246ssThan(nat,N,M2)) ) ) ).

% comm_monoid_set.atLeastLessThan_rev
tff(fact_7353_comm__monoid__set_OatLeastAtMost__rev,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat,M2: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,N,M2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ajh(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M2)),set_or1337092689740270186AtMost(nat,N,M2)) ) ) ).

% comm_monoid_set.atLeastAtMost_rev
tff(fact_7354_comm__monoid__set_Onat__group,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),K2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,A),fun(nat,fun(nat,A))),aTP_Lamp_aji(fun(A,fun(A,A)),fun(A,fun(fun(nat,A),fun(nat,fun(nat,A)))),F3),Z2),G3),K2)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K2))) ) ) ).

% comm_monoid_set.nat_group
tff(fact_7355_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat,M2: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,N,M2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ajh(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),N),M2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M2)) ) ) ).

% comm_monoid_set.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7356_comm__monoid__set_Oin__pairs,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2),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(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,A),fun(nat,A),aTP_Lamp_ajf(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),F3),G3)),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ).

% comm_monoid_set.in_pairs
tff(fact_7357_comm__monoid__set_Oub__add__nat,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,M2: nat,N: nat,G3: fun(nat,A),P: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P))) = aa(A,A,aa(A,fun(A,A),F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P)))) ) ) ) ).

% comm_monoid_set.ub_add_nat
tff(fact_7358_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,M2,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M2))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ).

% comm_monoid_set.atLeastLessThan_shift_0
tff(fact_7359_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,M2: nat,N: nat,G3: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N))
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M2))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M2))) ) ) ) ).

% comm_monoid_set.atLeastAtMost_shift_0
tff(fact_7360_comm__monoid__set_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),M2: nat,K2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ).

% comm_monoid_set.atLeastAtMost_shift_bounds
tff(fact_7361_comm__monoid__set_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),M2: nat,K2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or7035219750837199246ssThan(nat,M2,N)) ) ) ).

% comm_monoid_set.atLeastLessThan_shift_bounds
tff(fact_7362_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F3,aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ) ).

% comm_monoid_set.atLeast0_lessThan_Suc_shift
tff(fact_7363_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F3,aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ) ).

% comm_monoid_set.atLeast0_atMost_Suc_shift
tff(fact_7364_comm__monoid__set_OatLeastLessThan__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,fun(A,A)),Z2: A,H: fun(nat,B),M2: nat,N: nat,G3: fun(B,A)] :
          ( groups778175481326437816id_set(A,F3,Z2)
         => ( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M2,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,M2),aa(nat,B,H,N)))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F3,Z2),G3),set_or7035219750837199246ssThan(B,aa(nat,B,H,M2),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,B),fun(nat,A),comp(B,A,nat,G3),H)),set_or7035219750837199246ssThan(nat,M2,N)) ) ) ) ) ).

% comm_monoid_set.atLeastLessThan_reindex
tff(fact_7365_comm__monoid__set_OatLeastAtMost__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,fun(A,A)),Z2: A,H: fun(nat,B),M2: nat,N: nat,G3: fun(B,A)] :
          ( groups778175481326437816id_set(A,F3,Z2)
         => ( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M2,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,M2),aa(nat,B,H,N)))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F3,Z2),G3),set_or1337092689740270186AtMost(B,aa(nat,B,H,M2),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,B),fun(nat,A),comp(B,A,nat,G3),H)),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ) ) ).

% comm_monoid_set.atLeastAtMost_reindex
tff(fact_7366_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ).

% comm_monoid_set.atLeast_Suc_atMost_Suc_shift
tff(fact_7367_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,M2,N)) ) ) ).

% comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
tff(fact_7368_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_mw(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ).

% comm_monoid_set.atLeast_atMost_pred_shift
tff(fact_7369_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_mw(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),G3),set_or7035219750837199246ssThan(nat,M2,N)) ) ) ).

% comm_monoid_set.atLeast_lessThan_pred_shift
tff(fact_7370_comm__monoid__set_Otriangle__reindex,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,fun(nat,A)),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups_comm_monoid_F(A,product_prod(nat,nat),F3,Z2),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ig(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_ajk(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F3),Z2),G3)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).

% comm_monoid_set.triangle_reindex
tff(fact_7371_comm__monoid__set_Otriangle__reindex__eq,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(nat,fun(nat,A)),N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups_comm_monoid_F(A,product_prod(nat,nat),F3,Z2),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ib(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_ajk(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F3),Z2),G3)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% comm_monoid_set.triangle_reindex_eq
tff(fact_7372_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(int,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups_comm_monoid_F(A,int,F3,Z2),G3),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M2),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G3),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M2,N)) ) ) ).

% comm_monoid_set.atLeast_int_atMost_int_shift
tff(fact_7373_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,G3: fun(int,A),M2: nat,N: nat] :
      ( groups778175481326437816id_set(A,F3,Z2)
     => ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups_comm_monoid_F(A,int,F3,Z2),G3),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M2),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F3,Z2),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G3),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M2,N)) ) ) ).

% comm_monoid_set.atLeast_int_lessThan_int_shift
tff(fact_7374_max__nat_Omonoid__axioms,axiom,
    monoid(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.monoid_axioms
tff(fact_7375_module__hom__compose__scale,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [F3: fun(A,real),C3: B] :
          ( vector_linear(real,A,real,real_V8093663219630862766scaleR(A),times_times(real),F3)
         => real_Vector_linear(A,B,aa(B,fun(A,B),aTP_Lamp_ajl(fun(A,real),fun(B,fun(A,B)),F3),C3)) ) ) ).

% module_hom_compose_scale
tff(fact_7376_monoid_Oaxioms_I1_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A] :
      ( monoid(A,F3,Z2)
     => semigroup(A,F3) ) ).

% monoid.axioms(1)
tff(fact_7377_monoid_Oright__neutral,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,A3: A] :
      ( monoid(A,F3,Z2)
     => ( aa(A,A,aa(A,fun(A,A),F3,A3),Z2) = A3 ) ) ).

% monoid.right_neutral
tff(fact_7378_monoid_Oleft__neutral,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A,A3: A] :
      ( monoid(A,F3,Z2)
     => ( aa(A,A,aa(A,fun(A,A),F3,Z2),A3) = A3 ) ) ).

% monoid.left_neutral
tff(fact_7379_add_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => monoid(A,plus_plus(A),zero_zero(A)) ) ).

% add.monoid_axioms
tff(fact_7380_mult_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => monoid(A,times_times(A),one_one(A)) ) ).

% mult.monoid_axioms
tff(fact_7381_gcd__nat_Omonoid__axioms,axiom,
    monoid(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.monoid_axioms
tff(fact_7382_or_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.monoid_axioms
tff(fact_7383_xor_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).

% xor.monoid_axioms
tff(fact_7384_monoid_Ointro,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A] :
      ( semigroup(A,F3)
     => ( monoid_axioms(A,F3,Z2)
       => monoid(A,F3,Z2) ) ) ).

% monoid.intro
tff(fact_7385_monoid__def,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A] :
      ( monoid(A,F3,Z2)
    <=> ( semigroup(A,F3)
        & monoid_axioms(A,F3,Z2) ) ) ).

% monoid_def
tff(fact_7386_monoid__axioms_Ointro,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A] :
      ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F3,Z2),A5) = A5
     => ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F3,A5),Z2) = A5
       => monoid_axioms(A,F3,Z2) ) ) ).

% monoid_axioms.intro
tff(fact_7387_monoid__axioms__def,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A] :
      ( monoid_axioms(A,F3,Z2)
    <=> ( ! [A7: A] : aa(A,A,aa(A,fun(A,A),F3,Z2),A7) = A7
        & ! [A7: A] : aa(A,A,aa(A,fun(A,A),F3,A7),Z2) = A7 ) ) ).

% monoid_axioms_def
tff(fact_7388_monoid_Oaxioms_I2_J,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z2: A] :
      ( monoid(A,F3,Z2)
     => monoid_axioms(A,F3,Z2) ) ).

% monoid.axioms(2)
tff(fact_7389_Enum_Ortranclp__rtrancl__eq,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),X: A,Y: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),transitive_rtranclp(A,R),X),Y))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_rtrancl(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).

% Enum.rtranclp_rtrancl_eq
tff(fact_7390_rtrancl__def,axiom,
    ! [A: $tType,X5: set(product_prod(A,A))] : transitive_rtrancl(A,X5) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),transitive_rtranclp(A,aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),X5)))) ).

% rtrancl_def
tff(fact_7391_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X5: A,Xa: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),transitive_rtranclp(A,aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X5),Xa))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa)),transitive_rtrancl(A,R))) ) ).

% Transitive_Closure.rtranclp_rtrancl_eq
tff(fact_7392_converse__rtranclp__induct2,axiom,
    ! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Ax: A,Ay: B,Bx: A,By: B,P2: fun(A,fun(B,bool))] :
      ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By))
       => ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
              ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)))
             => ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P2,Ax),Ay)) ) ) ) ).

% converse_rtranclp_induct2
tff(fact_7393_converse__rtranclpE2,axiom,
    ! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Xa2: A,Xb: B,Za: A,Zb: B] :
      ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb)))
     => ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb) )
       => ~ ! [A5: A,B4: B] :
              ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)))
             => ~ pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))) ) ) ) ).

% converse_rtranclpE2
tff(fact_7394_rtranclp__induct2,axiom,
    ! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Ax: A,Ay: B,Bx: A,By: B,P2: fun(A,fun(B,bool))] :
      ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Ax),Ay))
       => ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
              ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)))
             => ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By)) ) ) ) ).

% rtranclp_induct2
tff(fact_7395_Card__order__infinite__not__under,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
       => ~ ? [A8: A] : field2(A,R) = order_under(A,R,A8) ) ) ).

% Card_order_infinite_not_under
tff(fact_7396_card__order__on__Card__order,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),R))
     => ( ( A6 = field2(A,R) )
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) ) ) ).

% card_order_on_Card_order
tff(fact_7397_card__order__on,axiom,
    ! [A: $tType,A6: set(A)] :
    ? [X_12: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),X_12)) ).

% card_order_on
tff(fact_7398_card__order__on__well__order__on,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),R))
     => order_well_order_on(A,A6,R) ) ).

% card_order_on_well_order_on
tff(fact_7399_natLeq__Card__order,axiom,
    pp(aa(set(product_prod(nat,nat)),bool,bNF_Ca8970107618336181345der_on(nat,field2(nat,bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq)) ).

% natLeq_Card_order
tff(fact_7400_infinite__Card__order__limit,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field2(A,R)))
         => ? [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),field2(A,R)))
              & ( A3 != X3 )
              & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3)),R)) ) ) ) ) ).

% infinite_Card_order_limit
tff(fact_7401_Card__order__trans,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Y: A,Z2: A] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ( X != Y )
       => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
         => ( ( Y != Z2 )
           => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R))
             => ( ( X != Z2 )
                & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z2)),R)) ) ) ) ) ) ) ).

% Card_order_trans
tff(fact_7402_exists__isCardSuc,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ? [X_12: set(product_prod(set(A),set(A)))] : pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca6246979054910435723ardSuc(A,R),X_12)) ) ).

% exists_isCardSuc
tff(fact_7403_cardSuc__finite,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R))))
      <=> pp(aa(set(A),bool,finite_finite2(A),field2(A,R))) ) ) ).

% cardSuc_finite
tff(fact_7404_cardSuc__isCardSuc,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca6246979054910435723ardSuc(A,R),bNF_Ca8387033319878233205ardSuc(A,R))) ) ).

% cardSuc_isCardSuc
tff(fact_7405_cardSuc__Card__order,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R))),bNF_Ca8387033319878233205ardSuc(A,R))) ) ).

% cardSuc_Card_order
tff(fact_7406_cardSuc__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : bNF_Ca8387033319878233205ardSuc(A,R) = fChoice(set(product_prod(set(A),set(A))),bNF_Ca6246979054910435723ardSuc(A,R)) ).

% cardSuc_def
tff(fact_7407_infinite__cardSuc__regularCard,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => bNF_Ca7133664381575040944arCard(set(A),bNF_Ca8387033319878233205ardSuc(A,R)) ) ) ).

% infinite_cardSuc_regularCard
tff(fact_7408_Cinfinite__limit__finite,axiom,
    ! [A: $tType,X6: set(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),field2(A,R)))
       => ( ( bNF_Ca4139267488887388095finite(A,R)
            & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
         => ? [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),field2(A,R)))
              & ! [Xa: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa),X6))
                 => ( ( Xa != X3 )
                    & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X3)),R)) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
tff(fact_7409_Cinfinite__limit,axiom,
    ! [A: $tType,X: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),field2(A,R)))
     => ( ( bNF_Ca4139267488887388095finite(A,R)
          & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
       => ? [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),field2(A,R)))
            & ( X != X3 )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X3)),R)) ) ) ) ).

% Cinfinite_limit
tff(fact_7410_Cinfinite__limit2,axiom,
    ! [A: $tType,X1: A,R: set(product_prod(A,A)),X2: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X1),field2(A,R)))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),field2(A,R)))
       => ( ( bNF_Ca4139267488887388095finite(A,R)
            & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
         => ? [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),field2(A,R)))
              & ( X1 != X3 )
              & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),X3)),R))
              & ( X2 != X3 )
              & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X3)),R)) ) ) ) ) ).

% Cinfinite_limit2
tff(fact_7411_card__of__cardSuc__finite,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,bNF_Ca6860139660246222851ard_of(A,A6)))))
    <=> pp(aa(set(A),bool,finite_finite2(A),A6)) ) ).

% card_of_cardSuc_finite
tff(fact_7412_cardSuc__UNION,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),As: fun(set(A),set(B)),B6: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
       => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R),As)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),image2(set(A),set(B),As,field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R))))))
           => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),R)),bNF_Wellorder_ordLeq(B,A)))
             => ? [X3: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R))))
                  & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(set(A),set(B),As,X3))) ) ) ) ) ) ) ).

% cardSuc_UNION
tff(fact_7413_card__of__Card__order,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Ca6860139660246222851ard_of(A,A6))) ).

% card_of_Card_order
tff(fact_7414_card__of__card__order__on,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),bNF_Ca6860139660246222851ard_of(A,A6))) ).

% card_of_card_order_on
tff(fact_7415_card__of__UNION__ordLeq__infinite__Field,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),I5: set(B),A6: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),R)),bNF_Wellorder_ordLeq(B,A)))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),I5))
               => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A6,X3))),R)),bNF_Wellorder_ordLeq(C,A))) )
           => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),image2(B,set(C),A6,I5)))),R)),bNF_Wellorder_ordLeq(C,A))) ) ) ) ) ).

% card_of_UNION_ordLeq_infinite_Field
tff(fact_7416_Card__order__Times__same__infinite,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
       => pp(aa(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,field2(A,R),aTP_Lamp_ajm(set(product_prod(A,A)),fun(A,set(A)),R)))),R)),bNF_Wellorder_ordLeq(product_prod(A,A),A))) ) ) ).

% Card_order_Times_same_infinite
tff(fact_7417_card__of__Sigma__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),I5: set(B),A6: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),R)),bNF_Wellorder_ordLeq(B,A)))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),I5))
               => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A6,X3))),R)),bNF_Wellorder_ordLeq(C,A))) )
           => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,I5,A6))),R)),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ) ).

% card_of_Sigma_ordLeq_infinite_Field
tff(fact_7418_card__of__Times__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),A6: set(B),B6: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),R)),bNF_Wellorder_ordLeq(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B6)),R)),bNF_Wellorder_ordLeq(C,A)))
         => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
           => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,A6,aTP_Lamp_adq(set(C),fun(B,set(C)),B6)))),R)),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ) ).

% card_of_Times_ordLeq_infinite_Field
tff(fact_7419_Card__order__iff__ordLeq__card__of,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Ca6860139660246222851ard_of(A,field2(A,R)))),bNF_Wellorder_ordLeq(A,A))) ) ).

% Card_order_iff_ordLeq_card_of
tff(fact_7420_cardSuc__mono__ordLeq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R4)),R4))
       => ( pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),bool,aa(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),fun(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),bool),member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),aa(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),bNF_Ca8387033319878233205ardSuc(A,R)),bNF_Ca8387033319878233205ardSuc(B,R4))),bNF_Wellorder_ordLeq(set(A),set(B))))
        <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordLeq(A,B))) ) ) ) ).

% cardSuc_mono_ordLeq
tff(fact_7421_card__order__on__def,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),R))
    <=> ( order_well_order_on(A,A6,R)
        & ! [R6: set(product_prod(A,A))] :
            ( order_well_order_on(A,A6,R6)
           => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R6)),bNF_Wellorder_ordLeq(A,A))) ) ) ) ).

% card_order_on_def
tff(fact_7422_card__of__empty1,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A))] :
      ( ( order_well_order_on(A,field2(A,R),R)
        | pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R)),bNF_Wellorder_ordLeq(B,A))) ) ).

% card_of_empty1
tff(fact_7423_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),A6: set(B),B6: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),R)),bNF_Wellorder_ordLeq(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),R)),bNF_Wellorder_ordLeq(B,A)))
         => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
           => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6))),R)),bNF_Wellorder_ordLeq(B,A))) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
tff(fact_7424_exists__minim__Card__order,axiom,
    ! [A: $tType,R2: set(set(product_prod(A,A)))] :
      ( ( R2 != bot_bot(set(set(product_prod(A,A)))) )
     => ( ! [X3: set(product_prod(A,A))] :
            ( pp(aa(set(set(product_prod(A,A))),bool,aa(set(product_prod(A,A)),fun(set(set(product_prod(A,A))),bool),member(set(product_prod(A,A))),X3),R2))
           => pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,X3)),X3)) )
       => ? [X3: set(product_prod(A,A))] :
            ( pp(aa(set(set(product_prod(A,A))),bool,aa(set(product_prod(A,A)),fun(set(set(product_prod(A,A))),bool),member(set(product_prod(A,A))),X3),R2))
            & ! [Xa: set(product_prod(A,A))] :
                ( pp(aa(set(set(product_prod(A,A))),bool,aa(set(product_prod(A,A)),fun(set(set(product_prod(A,A))),bool),member(set(product_prod(A,A))),Xa),R2))
               => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),X3),Xa)),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).

% exists_minim_Card_order
tff(fact_7425_Card__order__empty,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R)),bNF_Wellorder_ordLeq(B,A))) ) ).

% Card_order_empty
tff(fact_7426_Card__order__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),B2: B] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ( field2(A,R) != bot_bot(set(A)) )
       => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),insert(B,B2),bot_bot(set(B))))),R)),bNF_Wellorder_ordLeq(B,A))) ) ) ).

% Card_order_singl_ordLeq
tff(fact_7427_Card__order__Times2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),A6: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ( A6 != bot_bot(set(B)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),R),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,A6,aTP_Lamp_ajn(set(product_prod(A,A)),fun(B,set(A)),R))))),bNF_Wellorder_ordLeq(A,product_prod(B,A)))) ) ) ).

% Card_order_Times2
tff(fact_7428_Card__order__Times1,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),B6: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ( B6 != bot_bot(set(B)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),R),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R),aTP_Lamp_aci(set(B),fun(A,set(B)),B6))))),bNF_Wellorder_ordLeq(A,product_prod(A,B)))) ) ) ).

% Card_order_Times1
tff(fact_7429_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),A6: set(B),B6: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),R)),bNF_Wellorder_ordLeq(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B6)),R)),bNF_Wellorder_ordLeq(C,A)))
         => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
           => pp(aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A6,B6))),R)),bNF_Wellorder_ordLeq(sum_sum(B,C),A))) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
tff(fact_7430_Card__order__Plus1,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),B6: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),R),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,field2(A,R),B6)))),bNF_Wellorder_ordLeq(A,sum_sum(A,B)))) ) ).

% Card_order_Plus1
tff(fact_7431_Card__order__Plus2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),A6: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),R),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,A6,field2(A,R))))),bNF_Wellorder_ordLeq(A,sum_sum(B,A)))) ) ).

% Card_order_Plus2
tff(fact_7432_card__of__Plus__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A6: set(A),B6: set(B),C5: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool,aa(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),fun(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool),member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,C5,A6))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,C5,B6)))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B)))) ) ).

% card_of_Plus_mono2
tff(fact_7433_card__of__Plus__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A6: set(A),B6: set(B),C5: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool,aa(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),fun(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool),member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A6,C5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,B6,C5)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C)))) ) ).

% card_of_Plus_mono1
tff(fact_7434_card__of__Plus__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,A6: set(A),B6: set(B),C5: set(C),D5: set(D)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),bNF_Ca6860139660246222851ard_of(C,C5)),bNF_Ca6860139660246222851ard_of(D,D5))),bNF_Wellorder_ordLeq(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,aa(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),fun(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool),member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A6,C5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,B6,D5)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% card_of_Plus_mono
tff(fact_7435_card__of__Plus2,axiom,
    ! [B: $tType,A: $tType,B6: set(A),A6: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(A,B6)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,A6,B6)))),bNF_Wellorder_ordLeq(A,sum_sum(B,A)))) ).

% card_of_Plus2
tff(fact_7436_card__of__Plus1,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,B6)))),bNF_Wellorder_ordLeq(A,sum_sum(A,B)))) ).

% card_of_Plus1
tff(fact_7437_card__of__Un__Plus__ordLeq,axiom,
    ! [A: $tType,A6: set(A),B6: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B6))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,A6,B6)))),bNF_Wellorder_ordLeq(A,sum_sum(A,A)))) ).

% card_of_Un_Plus_ordLeq
tff(fact_7438_infinite__iff__card__of__nat,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
    <=> pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(nat,nat)),fun(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(nat,top_top(set(nat)))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(nat,A))) ) ).

% infinite_iff_card_of_nat
tff(fact_7439_card__of__Times1,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( ( A6 != bot_bot(set(A)) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A))))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(B,B6)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B6,aTP_Lamp_acm(set(A),fun(B,set(A)),A6))))),bNF_Wellorder_ordLeq(B,product_prod(B,A)))) ) ).

% card_of_Times1
tff(fact_7440_card__of__Times2,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( ( A6 != bot_bot(set(A)) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(B,B6)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6))))),bNF_Wellorder_ordLeq(B,product_prod(A,B)))) ) ).

% card_of_Times2
tff(fact_7441_card__of__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B2: B] :
      ( ( A6 != bot_bot(set(A)) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),insert(B,B2),bot_bot(set(B))))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A))) ) ).

% card_of_singl_ordLeq
tff(fact_7442_card__of__empty,axiom,
    ! [B: $tType,A: $tType,A6: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,A6))),bNF_Wellorder_ordLeq(A,B))) ).

% card_of_empty
tff(fact_7443_card__of__empty3,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B))))),bNF_Wellorder_ordLeq(A,B)))
     => ( A6 = bot_bot(set(A)) ) ) ).

% card_of_empty3
tff(fact_7444_card__of__least,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
      ( order_well_order_on(A,A6,R)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A6)),R)),bNF_Wellorder_ordLeq(A,A))) ) ).

% card_of_least
tff(fact_7445_card__of__ordLeq__infinite,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B)))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
       => ~ pp(aa(set(B),bool,finite_finite2(B),B6)) ) ) ).

% card_of_ordLeq_infinite
tff(fact_7446_card__of__ordLeq__finite,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(B),bool,finite_finite2(B),B6))
       => pp(aa(set(A),bool,finite_finite2(A),A6)) ) ) ).

% card_of_ordLeq_finite
tff(fact_7447_card__of__Sigma__ordLeq__infinite,axiom,
    ! [A: $tType,C: $tType,B: $tType,B6: set(A),I5: set(B),A6: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),B6))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),bNF_Ca6860139660246222851ard_of(A,B6))),bNF_Wellorder_ordLeq(B,A)))
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),I5))
             => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A6,X3))),bNF_Ca6860139660246222851ard_of(A,B6))),bNF_Wellorder_ordLeq(C,A))) )
         => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,I5,A6))),bNF_Ca6860139660246222851ard_of(A,B6))),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ).

% card_of_Sigma_ordLeq_infinite
tff(fact_7448_card__of__Times3,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A))))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,A6,aTP_Lamp_acv(set(A),fun(A,set(A)),A6))))),bNF_Wellorder_ordLeq(A,product_prod(A,A)))) ).

% card_of_Times3
tff(fact_7449_card__of__Sigma__mono1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A6: fun(A,set(B)),B6: fun(A,set(C))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),I5))
         => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),A6,X3))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),B6,X3)))),bNF_Wellorder_ordLeq(B,C))) )
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),bool,aa(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),fun(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),bool),member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,I5,A6))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,I5,B6)))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,C)))) ) ).

% card_of_Sigma_mono1
tff(fact_7450_card__of__Times__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A6: set(A),B6: set(B),C5: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),bool,aa(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),fun(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),bool),member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,A6,aTP_Lamp_adc(set(C),fun(A,set(C)),C5)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,B6,aTP_Lamp_adq(set(C),fun(B,set(C)),C5))))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).

% card_of_Times_mono1
tff(fact_7451_card__of__Times__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A6: set(A),B6: set(B),C5: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),bool,aa(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),fun(set(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),bool),member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Ca6860139660246222851ard_of(product_prod(C,A),product_Sigma(C,A,C5,aTP_Lamp_ajo(set(A),fun(C,set(A)),A6)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,C5,aTP_Lamp_ajp(set(B),fun(C,set(B)),B6))))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).

% card_of_Times_mono2
tff(fact_7452_infinite__iff__natLeq__ordLeq,axiom,
    ! [A: $tType,A6: set(A)] :
      ~ ( pp(aa(set(A),bool,finite_finite2(A),A6))
      <=> pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(nat,nat)),fun(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A))),bNF_Ca8665028551170535155natLeq),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(nat,A))) ) ).

% infinite_iff_natLeq_ordLeq
tff(fact_7453_card__of__UNION__Sigma,axiom,
    ! [B: $tType,A: $tType,A6: fun(B,set(A)),I5: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(B,set(A),A6,I5)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,I5,A6)))),bNF_Wellorder_ordLeq(A,product_prod(B,A)))) ).

% card_of_UNION_Sigma
tff(fact_7454_card__of__UNION__ordLeq__infinite,axiom,
    ! [B: $tType,A: $tType,C: $tType,B6: set(A),I5: set(B),A6: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),B6))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),bNF_Ca6860139660246222851ard_of(A,B6))),bNF_Wellorder_ordLeq(B,A)))
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),I5))
             => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A6,X3))),bNF_Ca6860139660246222851ard_of(A,B6))),bNF_Wellorder_ordLeq(C,A))) )
         => pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),image2(B,set(C),A6,I5)))),bNF_Ca6860139660246222851ard_of(A,B6))),bNF_Wellorder_ordLeq(C,A))) ) ) ) ).

% card_of_UNION_ordLeq_infinite
tff(fact_7455_card__of__ordLeq2,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
      ( ( A6 != bot_bot(set(A)) )
     => ( ? [G6: fun(B,A)] : image2(B,A,G6,B6) = A6
      <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B))) ) ) ).

% card_of_ordLeq2
tff(fact_7456_card__of__image,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A6: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,image2(B,A,F3,A6))),bNF_Ca6860139660246222851ard_of(B,A6))),bNF_Wellorder_ordLeq(A,B))) ).

% card_of_image
tff(fact_7457_card__of__ordLeqI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),B6: set(B)] :
      ( inj_on(A,B,F3,A6)
     => ( ! [A5: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A6))
           => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F3,A5)),B6)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B))) ) ) ).

% card_of_ordLeqI
tff(fact_7458_card__of__ordLeq,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
      ( ? [F5: fun(A,B)] :
          ( inj_on(A,B,F5,A6)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image2(A,B,F5,A6)),B6)) )
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B))) ) ).

% card_of_ordLeq
tff(fact_7459_surj__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,B6: set(A),F3: fun(B,A),A6: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),image2(B,A,F3,A6)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,B6)),bNF_Ca6860139660246222851ard_of(B,A6))),bNF_Wellorder_ordLeq(A,B))) ) ).

% surj_imp_ordLeq
tff(fact_7460_card__of__mono1,axiom,
    ! [A: $tType,A6: set(A),B6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(A,B6))),bNF_Wellorder_ordLeq(A,A))) ) ).

% card_of_mono1
tff(fact_7461_card__of__Plus__Times,axiom,
    ! [B: $tType,A: $tType,A12: A,A23: A,A6: set(A),B1: B,B22: B,B6: set(B)] :
      ( ( ( A12 != A23 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,A12),aa(set(A),set(A),insert(A,A23),bot_bot(set(A))))),A6)) )
     => ( ( ( B1 != B22 )
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),insert(B,B1),aa(set(B),set(B),insert(B,B22),bot_bot(set(B))))),B6)) )
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,aa(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),fun(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool),member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,B6))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6))))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B)))) ) ) ).

% card_of_Plus_Times
tff(fact_7462_card__of__Plus__Times__aux,axiom,
    ! [B: $tType,A: $tType,A12: A,A23: A,A6: set(A),B6: set(B)] :
      ( ( ( A12 != A23 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,A12),aa(set(A),set(A),insert(A,A23),bot_bot(set(A))))),A6)) )
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,aa(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),fun(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool),member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,B6))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6))))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B)))) ) ) ).

% card_of_Plus_Times_aux
tff(fact_7463_card__of__well__order__on,axiom,
    ! [A: $tType,A6: set(A)] : order_well_order_on(A,A6,bNF_Ca6860139660246222851ard_of(A,A6)) ).

% card_of_well_order_on
tff(fact_7464_Field__card__of,axiom,
    ! [A: $tType,A6: set(A)] : field2(A,bNF_Ca6860139660246222851ard_of(A,A6)) = A6 ).

% Field_card_of
tff(fact_7465_card__of__Well__order,axiom,
    ! [A: $tType,A6: set(A)] : order_well_order_on(A,field2(A,bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(A,A6)) ).

% card_of_Well_order
tff(fact_7466_ordLeq__Times__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),A6: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),bool,aa(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),fun(set(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),bool),member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Ca6860139660246222851ard_of(product_prod(C,A),product_Sigma(C,A,A6,aTP_Lamp_ajq(set(product_prod(A,A)),fun(C,set(A)),R)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,A6,aTP_Lamp_ajr(set(product_prod(B,B)),fun(C,set(B)),R4))))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).

% ordLeq_Times_mono2
tff(fact_7467_ordLeq__Times__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),C5: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),bool,aa(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),fun(set(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),bool),member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,field2(A,R),aTP_Lamp_adc(set(C),fun(A,set(C)),C5)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,field2(B,R4),aTP_Lamp_adq(set(C),fun(B,set(C)),C5))))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).

% ordLeq_Times_mono1
tff(fact_7468_card__of__mono2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),bNF_Ca6860139660246222851ard_of(B,field2(B,R4)))),bNF_Wellorder_ordLeq(A,B))) ) ).

% card_of_mono2
tff(fact_7469_card__of__Field__ordLess,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R),R)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),R)),bNF_Wellorder_ordLeq(A,A))) ) ).

% card_of_Field_ordLess
tff(fact_7470_ordLeq__Plus__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),P: set(product_prod(C,C)),P8: set(product_prod(D,D))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P),P8)),bNF_Wellorder_ordLeq(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,aa(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),fun(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool),member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R),field2(C,P)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,field2(B,R4),field2(D,P8))))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% ordLeq_Plus_mono
tff(fact_7471_ordLeq__Plus__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),C5: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool,aa(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),fun(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool),member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R),C5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,field2(B,R4),C5)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C)))) ) ).

% ordLeq_Plus_mono1
tff(fact_7472_ordLeq__Plus__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),A6: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordLeq(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool,aa(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),fun(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool),member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,A6,field2(A,R)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,A6,field2(B,R4))))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B)))) ) ).

% ordLeq_Plus_mono2
tff(fact_7473_cardSuc__ordLeq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),bNF_Ca8387033319878233205ardSuc(A,R))),bNF_Wellorder_ordLeq(A,set(A)))) ) ).

% cardSuc_ordLeq
tff(fact_7474_card__of__def,axiom,
    ! [A: $tType,A6: set(A)] : bNF_Ca6860139660246222851ard_of(A,A6) = fChoice(set(product_prod(A,A)),bNF_Ca8970107618336181345der_on(A,A6)) ).

% card_of_def
tff(fact_7475_regularCard__UNION,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),As: fun(A,set(B)),B6: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( bNF_Ca7133664381575040944arCard(A,R)
       => ( bNF_Ca3754400796208372196lChain(A,set(B),R,As)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),image2(A,set(B),As,field2(A,R)))))
           => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),R)),bNF_We4044943003108391690rdLess(B,A)))
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),field2(A,R)))
                  & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(A,set(B),As,X3))) ) ) ) ) ) ) ).

% regularCard_UNION
tff(fact_7476_Card__order__iff__Restr__underS,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),field2(A,R)))
           => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,order_underS(A,R,X4),aa(A,fun(A,set(A)),aTP_Lamp_ajs(set(product_prod(A,A)),fun(A,fun(A,set(A))),R),X4)))),bNF_Ca6860139660246222851ard_of(A,field2(A,R)))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).

% Card_order_iff_Restr_underS
tff(fact_7477_card__of__Plus__ordLess__infinite,axiom,
    ! [A: $tType,C: $tType,B: $tType,C5: set(A),A6: set(B),B6: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),C5))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),bNF_Ca6860139660246222851ard_of(A,C5))),bNF_We4044943003108391690rdLess(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B6)),bNF_Ca6860139660246222851ard_of(A,C5))),bNF_We4044943003108391690rdLess(C,A)))
         => pp(aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A6,B6))),bNF_Ca6860139660246222851ard_of(A,C5))),bNF_We4044943003108391690rdLess(sum_sum(B,C),A))) ) ) ) ).

% card_of_Plus_ordLess_infinite
tff(fact_7478_card__of__Pow,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(set(A),pow2(A,A6)))),bNF_We4044943003108391690rdLess(A,set(A)))) ).

% card_of_Pow
tff(fact_7479_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),A6: set(B),B6: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),R)),bNF_We4044943003108391690rdLess(B,A)))
         => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B6)),R)),bNF_We4044943003108391690rdLess(C,A)))
           => pp(aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A6,B6))),R)),bNF_We4044943003108391690rdLess(sum_sum(B,C),A))) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
tff(fact_7480_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_We4044943003108391690rdLess(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),R4)),bNF_We4044943003108391690rdLess(A,B))) ) ).

% BNF_Cardinal_Order_Relation.ordLess_Field
tff(fact_7481_Card__order__Pow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),bNF_Ca6860139660246222851ard_of(set(A),pow2(A,field2(A,R))))),bNF_We4044943003108391690rdLess(A,set(A)))) ) ).

% Card_order_Pow
tff(fact_7482_cardSuc__greater,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),bNF_Ca8387033319878233205ardSuc(A,R))),bNF_We4044943003108391690rdLess(A,set(A)))) ) ).

% cardSuc_greater
tff(fact_7483_finite__iff__ordLess__natLeq,axiom,
    ! [A: $tType,A6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A6))
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(A,A)),set(product_prod(nat,nat))),aa(set(product_prod(A,A)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(A,A)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(A,A)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca8665028551170535155natLeq)),bNF_We4044943003108391690rdLess(A,nat))) ) ).

% finite_iff_ordLess_natLeq
tff(fact_7484_card__of__ordLess2,axiom,
    ! [A: $tType,B: $tType,B6: set(A),A6: set(B)] :
      ( ( B6 != bot_bot(set(A)) )
     => ( ~ ? [F5: fun(B,A)] : image2(B,A,F5,A6) = B6
      <=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),bNF_Ca6860139660246222851ard_of(A,B6))),bNF_We4044943003108391690rdLess(B,A))) ) ) ).

% card_of_ordLess2
tff(fact_7485_cardSuc__least,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R4)),R4))
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_We4044943003108391690rdLess(A,B)))
         => pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(B,B))),bNF_Ca8387033319878233205ardSuc(A,R)),R4)),bNF_Wellorder_ordLeq(set(A),B))) ) ) ) ).

% cardSuc_least
tff(fact_7486_cardSuc__ordLess__ordLeq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R4)),R4))
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_We4044943003108391690rdLess(A,B)))
        <=> pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(B,B))),bNF_Ca8387033319878233205ardSuc(A,R)),R4)),bNF_Wellorder_ordLeq(set(A),B))) ) ) ) ).

% cardSuc_ordLess_ordLeq
tff(fact_7487_isCardSuc__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(set(A),set(A)))] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca6246979054910435723ardSuc(A,R),R4))
    <=> ( pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R4)),R4))
        & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),R4)),bNF_We4044943003108391690rdLess(A,set(A))))
        & ! [R7: set(product_prod(set(A),set(A)))] :
            ( ( pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R7)),R7))
              & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),R7)),bNF_We4044943003108391690rdLess(A,set(A)))) )
           => pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),bool,aa(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),fun(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),bool),member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),R4),R7)),bNF_Wellorder_ordLeq(set(A),set(A)))) ) ) ) ).

% isCardSuc_def
tff(fact_7488_card__of__ordLess,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( ~ ? [F5: fun(A,B)] :
            ( inj_on(A,B,F5,A6)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image2(A,B,F5,A6)),B6)) )
    <=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_We4044943003108391690rdLess(B,A))) ) ).

% card_of_ordLess
tff(fact_7489_cardSuc__ordLeq__ordLess,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R4)),R4))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A))))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A))))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A))))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),aa(set(product_prod(B,B)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),R4),bNF_Ca8387033319878233205ardSuc(A,R))),bNF_We4044943003108391690rdLess(B,set(A))))
        <=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),R)),bNF_Wellorder_ordLeq(B,A))) ) ) ) ).

% cardSuc_ordLeq_ordLess
tff(fact_7490_card__of__underS,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A3),field2(A,R)))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,order_underS(A,R,A3))),R)),bNF_We4044943003108391690rdLess(A,A))) ) ) ).

% card_of_underS
tff(fact_7491_cardSuc__least__aux,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(set(A),set(A)))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R4)),R4))
       => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),R4)),bNF_We4044943003108391690rdLess(A,set(A))))
         => pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),bool,aa(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),fun(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),bool),member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),bNF_Ca8387033319878233205ardSuc(A,R)),R4)),bNF_Wellorder_ordLeq(set(A),set(A)))) ) ) ) ).

% cardSuc_least_aux
tff(fact_7492_Card__order__Times__infinite,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),P: set(product_prod(B,B))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => ( ( field2(B,P) != bot_bot(set(B)) )
         => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P),R)),bNF_Wellorder_ordLeq(B,A)))
           => ( pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R),aTP_Lamp_ajt(set(product_prod(B,B)),fun(A,set(B)),P)))),R)),bNF_Wellorder_ordIso(product_prod(A,B),A)))
              & pp(aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,field2(B,P),aTP_Lamp_ajn(set(product_prod(A,A)),fun(B,set(A)),R)))),R)),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ) ) ).

% Card_order_Times_infinite
tff(fact_7493_card__of__Times__infinite__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( B6 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6))))),bNF_Wellorder_ordIso(A,product_prod(A,B)))) ) ) ) ).

% card_of_Times_infinite_simps(2)
tff(fact_7494_cardSuc__invar__ordIso,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R4)),R4))
       => ( pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),bool,aa(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),fun(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),bool),member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),aa(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),bNF_Ca8387033319878233205ardSuc(A,R)),bNF_Ca8387033319878233205ardSuc(B,R4))),bNF_Wellorder_ordIso(set(A),set(B))))
        <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordIso(A,B))) ) ) ) ).

% cardSuc_invar_ordIso
tff(fact_7495_card__order__on__ordIso,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A)),R4: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),R))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),R4))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R4)),bNF_Wellorder_ordIso(A,A))) ) ) ).

% card_order_on_ordIso
tff(fact_7496_card__of__Times__Plus__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,A6: set(A),B6: set(B),C5: set(C)] : pp(aa(set(product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))))),bool,aa(product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),fun(set(product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))))),bool),member(product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))))),aa(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),aa(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),fun(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))))),product_Pair(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),bNF_Ca6860139660246222851ard_of(product_prod(A,sum_sum(B,C)),product_Sigma(A,sum_sum(B,C),A6,aa(set(C),fun(A,set(sum_sum(B,C))),aTP_Lamp_aju(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),B6),C5)))),bNF_Ca6860139660246222851ard_of(sum_sum(product_prod(A,B),product_prod(A,C)),sum_Plus(product_prod(A,B),product_prod(A,C),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6)),product_Sigma(A,C,A6,aTP_Lamp_adc(set(C),fun(A,set(C)),C5)))))),bNF_Wellorder_ordIso(product_prod(A,sum_sum(B,C)),sum_sum(product_prod(A,B),product_prod(A,C))))) ).

% card_of_Times_Plus_distrib
tff(fact_7497_card__of__Plus__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A6: set(A),B6: set(B),C5: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool,aa(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),fun(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool),member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,C5,A6))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,C5,B6)))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B)))) ) ).

% card_of_Plus_cong2
tff(fact_7498_card__of__Plus__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A6: set(A),B6: set(B),C5: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool,aa(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),fun(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool),member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A6,C5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,B6,C5)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C)))) ) ).

% card_of_Plus_cong1
tff(fact_7499_card__of__Plus__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,A6: set(A),B6: set(B),C5: set(C),D5: set(D)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),bNF_Ca6860139660246222851ard_of(C,C5)),bNF_Ca6860139660246222851ard_of(D,D5))),bNF_Wellorder_ordIso(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,aa(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),fun(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool),member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A6,C5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,B6,D5)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% card_of_Plus_cong
tff(fact_7500_card__of__Plus__Times__bool,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool))))),bool,aa(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool)))),fun(set(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool))))),bool),member(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool))))),aa(set(product_prod(product_prod(A,bool),product_prod(A,bool))),product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),fun(set(product_prod(product_prod(A,bool),product_prod(A,bool))),product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool))))),product_Pair(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,A6,A6))),bNF_Ca6860139660246222851ard_of(product_prod(A,bool),product_Sigma(A,bool,A6,aTP_Lamp_ajv(A,set(bool)))))),bNF_Wellorder_ordIso(sum_sum(A,A),product_prod(A,bool)))) ).

% card_of_Plus_Times_bool
tff(fact_7501_card__of__empty2,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B))))),bNF_Wellorder_ordIso(A,B)))
     => ( A6 = bot_bot(set(A)) ) ) ).

% card_of_empty2
tff(fact_7502_card__of__empty__ordIso,axiom,
    ! [B: $tType,A: $tType] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B))))),bNF_Wellorder_ordIso(A,B))) ).

% card_of_empty_ordIso
tff(fact_7503_card__of__ordIso__finite,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(A),bool,finite_finite2(A),A6))
      <=> pp(aa(set(B),bool,finite_finite2(B),B6)) ) ) ).

% card_of_ordIso_finite
tff(fact_7504_card__of__refl,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(A,A))) ).

% card_of_refl
tff(fact_7505_ordIso__Plus__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),P: set(product_prod(C,C)),P8: set(product_prod(D,D))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool,aa(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),fun(set(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),bool),member(product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P),P8)),bNF_Wellorder_ordIso(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,aa(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),fun(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool),member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R),field2(C,P)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,field2(B,R4),field2(D,P8))))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% ordIso_Plus_cong
tff(fact_7506_ordIso__Plus__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),C5: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool,aa(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),fun(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),bool),member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R),C5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,field2(B,R4),C5)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C)))) ) ).

% ordIso_Plus_cong1
tff(fact_7507_ordIso__Plus__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),A6: set(C)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool,aa(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),fun(set(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),bool),member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,A6,field2(A,R)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,A6,field2(B,R4))))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B)))) ) ).

% ordIso_Plus_cong2
tff(fact_7508_card__of__cong,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),bNF_Ca6860139660246222851ard_of(B,field2(B,R4)))),bNF_Wellorder_ordIso(A,B))) ) ).

% card_of_cong
tff(fact_7509_Card__order__ordIso2,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R4)),bNF_Wellorder_ordIso(A,B)))
       => pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R4)),R4)) ) ) ).

% Card_order_ordIso2
tff(fact_7510_Card__order__ordIso,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R4),R)),bNF_Wellorder_ordIso(B,A)))
       => pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R4)),R4)) ) ) ).

% Card_order_ordIso
tff(fact_7511_card__of__unique,axiom,
    ! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A6),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(A,A))) ) ).

% card_of_unique
tff(fact_7512_card__of__ordIso,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
      ( ? [F5: fun(A,B)] : bij_betw(A,B,F5,A6,B6)
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordIso(A,B))) ) ).

% card_of_ordIso
tff(fact_7513_card__of__ordIsoI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A6: set(A),B6: set(B)] :
      ( bij_betw(A,B,F3,A6,B6)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordIso(A,B))) ) ).

% card_of_ordIsoI
tff(fact_7514_card__of__Times__commute,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] : pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,aa(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),fun(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool),member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B6,aTP_Lamp_acm(set(A),fun(B,set(A)),A6))))),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A)))) ).

% card_of_Times_commute
tff(fact_7515_internalize__card__of__ordLeq2,axiom,
    ! [A: $tType,B: $tType,A6: set(A),C5: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,C5))),bNF_Wellorder_ordLeq(A,B)))
    <=> ? [B8: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B8),C5))
          & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B8))),bNF_Wellorder_ordIso(A,B)))
          & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(B,B8)),bNF_Ca6860139660246222851ard_of(B,C5))),bNF_Wellorder_ordLeq(B,B))) ) ) ).

% internalize_card_of_ordLeq2
tff(fact_7516_ordIso__card__of__imp__Card__order,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),A6: set(B)] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),bNF_Ca6860139660246222851ard_of(B,A6))),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) ) ).

% ordIso_card_of_imp_Card_order
tff(fact_7517_Card__order__iff__ordIso__card__of,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Ca6860139660246222851ard_of(A,field2(A,R)))),bNF_Wellorder_ordIso(A,A))) ) ).

% Card_order_iff_ordIso_card_of
tff(fact_7518_card__of__Field__ordIso,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),R)),bNF_Wellorder_ordIso(A,A))) ) ).

% card_of_Field_ordIso
tff(fact_7519_internalize__card__of__ordLeq,axiom,
    ! [A: $tType,B: $tType,A6: set(A),R: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R)),bNF_Wellorder_ordLeq(A,B)))
    <=> ? [B8: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B8),field2(B,R)))
          & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B8))),bNF_Wellorder_ordIso(A,B)))
          & pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(B,B8)),R)),bNF_Wellorder_ordLeq(B,B))) ) ) ).

% internalize_card_of_ordLeq
tff(fact_7520_card__of__ordIso__finite__Field,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),A6: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),bNF_Ca6860139660246222851ard_of(B,A6))),bNF_Wellorder_ordIso(A,B)))
       => ( pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
        <=> pp(aa(set(B),bool,finite_finite2(B),A6)) ) ) ) ).

% card_of_ordIso_finite_Field
tff(fact_7521_card__of__Times__same__infinite,axiom,
    ! [A: $tType,A6: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => pp(aa(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,A6,aTP_Lamp_acv(set(A),fun(A,set(A)),A6)))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(product_prod(A,A),A))) ) ).

% card_of_Times_same_infinite
tff(fact_7522_regularCard__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( bNF_Ca7133664381575040944arCard(A,R)
    <=> ! [K6: set(A)] :
          ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),K6),field2(A,R)))
            & bNF_Ca7293521722713021262ofinal(A,K6,R) )
         => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,K6)),R)),bNF_Wellorder_ordIso(A,A))) ) ) ).

% regularCard_def
tff(fact_7523_card__of__Times__infinite__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( B6 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B6,aTP_Lamp_acm(set(A),fun(B,set(A)),A6)))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ).

% card_of_Times_infinite_simps(3)
tff(fact_7524_card__of__Times__infinite__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( B6 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6)))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(product_prod(A,B),A))) ) ) ) ).

% card_of_Times_infinite_simps(1)
tff(fact_7525_card__of__Times__infinite,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( B6 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
         => ( pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6)))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(product_prod(A,B),A)))
            & pp(aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B6,aTP_Lamp_acm(set(A),fun(B,set(A)),A6)))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ) ).

% card_of_Times_infinite
tff(fact_7526_card__of__Times__infinite__simps_I4_J,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( ( B6 != bot_bot(set(B)) )
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B6,aTP_Lamp_acm(set(A),fun(B,set(A)),A6))))),bNF_Wellorder_ordIso(A,product_prod(B,A)))) ) ) ) ).

% card_of_Times_infinite_simps(4)
tff(fact_7527_Card__order__Plus__infinite,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),P: set(product_prod(B,B))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P),R)),bNF_Wellorder_ordLeq(B,A)))
         => ( pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,field2(A,R),field2(B,P)))),R)),bNF_Wellorder_ordIso(sum_sum(A,B),A)))
            & pp(aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,field2(B,P),field2(A,R)))),R)),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ) ) ).

% Card_order_Plus_infinite
tff(fact_7528_card__of__Plus__infinite,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
       => ( pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,B6))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(sum_sum(A,B),A)))
          & pp(aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B6,A6))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ) ).

% card_of_Plus_infinite
tff(fact_7529_card__of__Plus__assoc,axiom,
    ! [C: $tType,B: $tType,A: $tType,A6: set(A),B6: set(B),C5: set(C)] : pp(aa(set(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),bool,aa(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),fun(set(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),bool),member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),aa(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),fun(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),bNF_Ca6860139660246222851ard_of(sum_sum(sum_sum(A,B),C),sum_Plus(sum_sum(A,B),C,sum_Plus(A,B,A6,B6),C5))),bNF_Ca6860139660246222851ard_of(sum_sum(A,sum_sum(B,C)),sum_Plus(A,sum_sum(B,C),A6,sum_Plus(B,C,B6,C5))))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),C),sum_sum(A,sum_sum(B,C))))) ).

% card_of_Plus_assoc
tff(fact_7530_card__of__bool,axiom,
    ! [A: $tType,A12: A,A23: A] :
      ( ( A12 != A23 )
     => pp(aa(set(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(bool,top_top(set(bool)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),insert(A,A12),aa(set(A),set(A),insert(A,A23),bot_bot(set(A))))))),bNF_Wellorder_ordIso(bool,A))) ) ).

% card_of_bool
tff(fact_7531_card__of__Plus__commute,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] : pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,aa(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),fun(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool),member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,B6))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B6,A6)))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(B,A)))) ).

% card_of_Plus_commute
tff(fact_7532_card__of__nat,axiom,
    pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(nat,top_top(set(nat)))),bNF_Ca8665028551170535155natLeq)),bNF_Wellorder_ordIso(nat,nat))) ).

% card_of_nat
tff(fact_7533_card__of__Field__natLeq,axiom,
    pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,aa(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool),member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(nat,field2(nat,bNF_Ca8665028551170535155natLeq))),bNF_Ca8665028551170535155natLeq)),bNF_Wellorder_ordIso(nat,nat))) ).

% card_of_Field_natLeq
tff(fact_7534_card__of__Plus__empty1,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,bot_bot(set(B)))))),bNF_Wellorder_ordIso(A,sum_sum(A,B)))) ).

% card_of_Plus_empty1
tff(fact_7535_card__of__Plus__empty2,axiom,
    ! [B: $tType,A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,aa(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool),member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,bot_bot(set(B)),A6)))),bNF_Wellorder_ordIso(A,sum_sum(B,A)))) ).

% card_of_Plus_empty2
tff(fact_7536_card__of__Plus__infinite2,axiom,
    ! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B6,A6))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ).

% card_of_Plus_infinite2
tff(fact_7537_card__of__Plus__infinite1,axiom,
    ! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A6))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordLeq(B,A)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),bool,aa(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),fun(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),bool),member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A6,B6))),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_Wellorder_ordIso(sum_sum(A,B),A))) ) ) ).

% card_of_Plus_infinite1
tff(fact_7538_card__of__Pow__Func,axiom,
    ! [A: $tType,A6: set(A)] : pp(aa(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool))))),bool,aa(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool)))),fun(set(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool))))),bool),member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool))))),aa(set(product_prod(fun(A,bool),fun(A,bool))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(fun(A,bool),fun(A,bool))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool)))),bNF_Ca6860139660246222851ard_of(set(A),pow2(A,A6))),bNF_Ca6860139660246222851ard_of(fun(A,bool),bNF_Wellorder_Func(A,bool,A6,top_top(set(bool)))))),bNF_Wellorder_ordIso(set(A),fun(A,bool)))) ).

% card_of_Pow_Func
tff(fact_7539_card__of__Func__UNIV,axiom,
    ! [B: $tType,A: $tType,B6: set(B)] : pp(aa(set(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),bool,aa(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),fun(set(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),bool),member(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),aa(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),fun(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),product_Pair(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),bNF_Ca6860139660246222851ard_of(fun(A,B),bNF_Wellorder_Func(A,B,top_top(set(A)),B6))),bNF_Ca6860139660246222851ard_of(fun(A,B),aa(fun(fun(A,B),bool),set(fun(A,B)),collect(fun(A,B)),aTP_Lamp_ajw(set(B),fun(fun(A,B),bool),B6))))),bNF_Wellorder_ordIso(fun(A,B),fun(A,B)))) ).

% card_of_Func_UNIV
tff(fact_7540_Func__Times__Range,axiom,
    ! [C: $tType,B: $tType,A: $tType,A6: set(A),B6: set(B),C5: set(C)] : pp(aa(set(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),bool,aa(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),fun(set(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),bool),member(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),aa(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),fun(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),product_Pair(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),bNF_Ca6860139660246222851ard_of(fun(A,product_prod(B,C)),bNF_Wellorder_Func(A,product_prod(B,C),A6,product_Sigma(B,C,B6,aTP_Lamp_adq(set(C),fun(B,set(C)),C5))))),bNF_Ca6860139660246222851ard_of(product_prod(fun(A,B),fun(A,C)),product_Sigma(fun(A,B),fun(A,C),bNF_Wellorder_Func(A,B,A6,B6),aa(set(C),fun(fun(A,B),set(fun(A,C))),aTP_Lamp_ajx(set(A),fun(set(C),fun(fun(A,B),set(fun(A,C)))),A6),C5))))),bNF_Wellorder_ordIso(fun(A,product_prod(B,C)),product_prod(fun(A,B),fun(A,C))))) ).

% Func_Times_Range
tff(fact_7541_card__of__Func__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,A6: set(A),B6: set(B),C5: set(C)] : pp(aa(set(product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))))),bool,aa(product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),fun(set(product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))))),bool),member(product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))))),aa(set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))),product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),aa(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),fun(set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))),product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))))),product_Pair(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),bNF_Ca6860139660246222851ard_of(fun(product_prod(A,B),C),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A6,aTP_Lamp_aci(set(B),fun(A,set(B)),B6)),C5))),bNF_Ca6860139660246222851ard_of(fun(A,fun(B,C)),bNF_Wellorder_Func(A,fun(B,C),A6,bNF_Wellorder_Func(B,C,B6,C5))))),bNF_Wellorder_ordIso(fun(product_prod(A,B),C),fun(A,fun(B,C))))) ).

% card_of_Func_Times
tff(fact_7542_ATP_Olambda__1,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_ahd(nat,set(old_node(A,product_unit))),Uu) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_aha(nat,set(old_node(A,product_unit))),aTP_Lamp_ahc(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,Uu)) ) ).

% ATP.lambda_1
tff(fact_7543_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_agz(nat,set(old_node(A,product_unit))),Uu) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_agx(nat,set(old_node(A,product_unit))),aTP_Lamp_agy(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,Uu)) ) ).

% ATP.lambda_2
tff(fact_7544_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_gi(nat,real),Uu) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uu)),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_3
tff(fact_7545_ATP_Olambda__4,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ta(A,A),Uu) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),exp(A,Uu)),one_one(A)),Uu) ) ).

% ATP.lambda_4
tff(fact_7546_ATP_Olambda__5,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_xk(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(set(A)),nat,finite_card(set(A)),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uu)))) ).

% ATP.lambda_5
tff(fact_7547_ATP_Olambda__6,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_aae(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_6
tff(fact_7548_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_aez(A,bool),Uu))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uu),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Uu)) ) ) ) ).

% ATP.lambda_7
tff(fact_7549_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_by(nat,real),Uu) = aa(nat,real,aa(real,fun(nat,real),power_power(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_8
tff(fact_7550_ATP_Olambda__9,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_ip(real,bool),Uu))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Uu))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uu),aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( cos(real,Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_9
tff(fact_7551_ATP_Olambda__10,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_tz(nat,A),Uu) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu)),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_10
tff(fact_7552_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa(nat,int,aTP_Lamp_agv(nat,int),Uu) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Uu))),one_one(int)) ).

% ATP.lambda_11
tff(fact_7553_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_gb(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uu)),aa(nat,real,aa(real,fun(nat,real),power_power(real),zero_zero(real)),Uu)) ).

% ATP.lambda_12
tff(fact_7554_ATP_Olambda__13,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_ty(nat,A),Uu) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uu),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))) ) ).

% ATP.lambda_13
tff(fact_7555_ATP_Olambda__14,axiom,
    ! [A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: product_prod(A,A)] : aa(product_prod(A,A),A,aTP_Lamp_abz(product_prod(A,A),A),Uu) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(product_prod(A,A),A,product_fst(A,A),Uu)),aa(product_prod(A,A),A,product_snd(A,A),Uu)) ) ).

% ATP.lambda_14
tff(fact_7556_ATP_Olambda__15,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),product_prod(nat,list(A)),aTP_Lamp_afa(list(A),product_prod(nat,list(A))),Uu) = aa(list(A),product_prod(nat,list(A)),aa(nat,fun(list(A),product_prod(nat,list(A))),product_Pair(nat,list(A)),aa(list(A),nat,size_size(list(A)),Uu)),Uu) ).

% ATP.lambda_15
tff(fact_7557_ATP_Olambda__16,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( pp(aa(nat,bool,aTP_Lamp_ke(nat,bool),Uu))
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_16
tff(fact_7558_ATP_Olambda__17,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_mw(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_17
tff(fact_7559_ATP_Olambda__18,axiom,
    ! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_aan(B,product_prod(B,B)),Uu) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),Uu) ).

% ATP.lambda_18
tff(fact_7560_ATP_Olambda__19,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_mq(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu) ).

% ATP.lambda_19
tff(fact_7561_ATP_Olambda__20,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_cw(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_20
tff(fact_7562_ATP_Olambda__21,axiom,
    ! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_aee(nat,product_prod(nat,nat)),Uu) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uu),zero_zero(nat)) ).

% ATP.lambda_21
tff(fact_7563_ATP_Olambda__22,axiom,
    ! [B: $tType,Uu: option(B)] :
      ( pp(aa(option(B),bool,aTP_Lamp_aho(option(B),bool),Uu))
    <=> ( Uu = none(B) ) ) ).

% ATP.lambda_22
tff(fact_7564_ATP_Olambda__23,axiom,
    ! [A: $tType,Uu: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_ahn(option(A),bool),Uu))
    <=> ( Uu = none(A) ) ) ).

% ATP.lambda_23
tff(fact_7565_ATP_Olambda__24,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_ahc(nat,set(old_node(A,product_unit))),Uu) = aa(product_prod(nat,nat),set(old_node(A,product_unit)),aa(fun(nat,fun(nat,set(old_node(A,product_unit)))),fun(product_prod(nat,nat),set(old_node(A,product_unit))),product_case_prod(nat,nat,set(old_node(A,product_unit))),aTP_Lamp_ahb(nat,fun(nat,set(old_node(A,product_unit))))),aa(nat,product_prod(nat,nat),nat_prod_decode,Uu)) ) ).

% ATP.lambda_24
tff(fact_7566_ATP_Olambda__25,axiom,
    ! [Uu: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_aeb(product_prod(int,int),bool),Uu))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uu)))) ) ).

% ATP.lambda_25
tff(fact_7567_ATP_Olambda__26,axiom,
    ! [Uu: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_aii(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aTP_Lamp_aih(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu)) ).

% ATP.lambda_26
tff(fact_7568_ATP_Olambda__27,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_tx(nat,A),Uu) = divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_27
tff(fact_7569_ATP_Olambda__28,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_agw(nat,nat),Uu) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)) ).

% ATP.lambda_28
tff(fact_7570_ATP_Olambda__29,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_nu(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Uu)),aa(nat,nat,suc,zero_zero(nat)))) ).

% ATP.lambda_29
tff(fact_7571_ATP_Olambda__30,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_aad(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).

% ATP.lambda_30
tff(fact_7572_ATP_Olambda__31,axiom,
    ! [B: $tType,Uu: option(B)] :
      ( pp(aa(option(B),bool,aTP_Lamp_ahq(option(B),bool),Uu))
    <=> ( Uu != none(B) ) ) ).

% ATP.lambda_31
tff(fact_7573_ATP_Olambda__32,axiom,
    ! [A: $tType,Uu: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_ahp(option(A),bool),Uu))
    <=> ( Uu != none(A) ) ) ).

% ATP.lambda_32
tff(fact_7574_ATP_Olambda__33,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( pp(aa(list(B),bool,aTP_Lamp_nv(list(B),bool),Uu))
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_33
tff(fact_7575_ATP_Olambda__34,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_nr(list(A),bool),Uu))
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_34
tff(fact_7576_ATP_Olambda__35,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: A] : aa(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_acb(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),Uu) = aa(fun(B,fun(C,product_prod(product_prod(A,B),C))),fun(product_prod(B,C),product_prod(product_prod(A,B),C)),product_case_prod(B,C,product_prod(product_prod(A,B),C)),aTP_Lamp_aca(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu)) ).

% ATP.lambda_35
tff(fact_7577_ATP_Olambda__36,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B))] : aa(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))),aTP_Lamp_abh(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),Uu) = aa(fun(A,fun(B,fun(A,option(B)))),fun(product_prod(A,B),fun(A,option(B))),product_case_prod(A,B,fun(A,option(B))),aTP_Lamp_abg(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu)) ).

% ATP.lambda_36
tff(fact_7578_ATP_Olambda__37,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: B] : aa(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_mu(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu) = aa(fun(A,fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(A,C),product_prod(A,product_prod(B,C))),product_case_prod(A,C,product_prod(A,product_prod(B,C))),aTP_Lamp_mt(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).

% ATP.lambda_37
tff(fact_7579_ATP_Olambda__38,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_od(real,real),Uu) = suminf(real,aTP_Lamp_gh(real,fun(nat,real),Uu)) ).

% ATP.lambda_38
tff(fact_7580_ATP_Olambda__39,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_xz(nat,set(nat)),Uu) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_bo(nat,fun(nat,bool),Uu)) ).

% ATP.lambda_39
tff(fact_7581_ATP_Olambda__40,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_tv(nat,real),Uu) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu))) ).

% ATP.lambda_40
tff(fact_7582_ATP_Olambda__41,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_tp(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_41
tff(fact_7583_ATP_Olambda__42,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_nt(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(B),nat,size_size(list(B)),Uu)) ).

% ATP.lambda_42
tff(fact_7584_ATP_Olambda__43,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_nq(list(A),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Uu)) ).

% ATP.lambda_43
tff(fact_7585_ATP_Olambda__44,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_zc(num,option(num)),Uu) = aa(num,option(num),some(num),bit0(Uu)) ).

% ATP.lambda_44
tff(fact_7586_ATP_Olambda__45,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_ia(nat,fun(nat,product_prod(nat,nat))),Uu) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_45
tff(fact_7587_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_aha(nat,set(old_node(A,product_unit))),Uu) = old_Leaf(A,product_unit,from_nat(A,Uu)) ) ).

% ATP.lambda_46
tff(fact_7588_ATP_Olambda__47,axiom,
    ! [Uu: nat] : aa(nat,extended_enat,aTP_Lamp_aex(nat,extended_enat),Uu) = extended_enat2(aa(nat,nat,suc,Uu)) ).

% ATP.lambda_47
tff(fact_7589_ATP_Olambda__48,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_agy(nat,set(old_node(A,product_unit))),Uu) = old_In1(A,product_unit,nth_item(A,Uu)) ) ).

% ATP.lambda_48
tff(fact_7590_ATP_Olambda__49,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_agx(nat,set(old_node(A,product_unit))),Uu) = old_In0(A,product_unit,nth_item(A,Uu)) ) ).

% ATP.lambda_49
tff(fact_7591_ATP_Olambda__50,axiom,
    ! [Uu: fun(nat,rat)] :
      ( pp(aa(fun(nat,rat),bool,aTP_Lamp_afd(fun(nat,rat),bool),Uu))
    <=> ? [R5: rat] :
          ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R5))
          & ? [K3: nat] :
            ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N5))
             => pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R5),aa(nat,rat,Uu,N5))) ) ) ) ).

% ATP.lambda_50
tff(fact_7592_ATP_Olambda__51,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_aiu(product_prod(A,A),bool),Uu))
        <=> ? [X4: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ) ).

% ATP.lambda_51
tff(fact_7593_ATP_Olambda__52,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aTP_Lamp_aau(product_prod(A,A),bool),Uu))
    <=> ? [X4: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ).

% ATP.lambda_52
tff(fact_7594_ATP_Olambda__53,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_aam(real,bool),Uu))
    <=> ? [I: int,N5: nat] :
          ( ( Uu = divide_divide(real,aa(int,real,ring_1_of_int(real),I),aa(nat,real,semiring_1_of_nat(real),N5)) )
          & ( N5 != zero_zero(nat) ) ) ) ).

% ATP.lambda_53
tff(fact_7595_ATP_Olambda__54,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_aiv(product_prod(A,A),bool),Uu))
        <=> ? [X4: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y5)) ) ) ) ).

% ATP.lambda_54
tff(fact_7596_ATP_Olambda__55,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_aiw(product_prod(A,A),bool),Uu))
        <=> ? [X4: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y5),X4)) ) ) ) ).

% ATP.lambda_55
tff(fact_7597_ATP_Olambda__56,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_yg(product_prod(A,A),bool),Uu))
        <=> ? [X4: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5)) ) ) ) ).

% ATP.lambda_56
tff(fact_7598_ATP_Olambda__57,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_yh(product_prod(A,A),bool),Uu))
        <=> ? [X4: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X4)) ) ) ) ).

% ATP.lambda_57
tff(fact_7599_ATP_Olambda__58,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_yc(product_prod(A,A),bool),Uu))
        <=> ? [X4: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5) )
              & ( X4 != Y5 ) ) ) ) ).

% ATP.lambda_58
tff(fact_7600_ATP_Olambda__59,axiom,
    ! [A: $tType,B: $tType,Uu: product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))] :
      ( pp(aa(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool,aTP_Lamp_agq(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool),Uu))
    <=> ? [F5: fun(nat,sum_sum(B,nat)),X4: sum_sum(A,nat),K3: nat] :
          ( ( Uu = aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),F5),X4) )
          & ( aa(nat,sum_sum(B,nat),F5,K3) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ) ) ) ).

% ATP.lambda_59
tff(fact_7601_ATP_Olambda__60,axiom,
    ! [B: $tType,Uu: nat] : aa(nat,sum_sum(B,nat),aTP_Lamp_agu(nat,sum_sum(B,nat)),Uu) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ).

% ATP.lambda_60
tff(fact_7602_ATP_Olambda__61,axiom,
    ! [A: $tType,Uu: nat] : aa(nat,sum_sum(A,nat),aTP_Lamp_agr(nat,sum_sum(A,nat)),Uu) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ).

% ATP.lambda_61
tff(fact_7603_ATP_Olambda__62,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hs(A,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_62
tff(fact_7604_ATP_Olambda__63,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dx(nat,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_63
tff(fact_7605_ATP_Olambda__64,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hr(A,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_64
tff(fact_7606_ATP_Olambda__65,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: product_prod(A,C),Uua: product_prod(C,B)] : aa(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_zf(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = if(list(product_prod(A,B)),aa(C,bool,aa(C,fun(C,bool),fequal(C),aa(product_prod(A,C),C,product_snd(A,C),Uu)),aa(product_prod(C,B),C,product_fst(C,B),Uua)),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Uu)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).

% ATP.lambda_65
tff(fact_7607_ATP_Olambda__66,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_kd(code_integer,fun(code_integer,int)),Uu),Uua) = if(int,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),code_int_of_integer(Uu)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),code_int_of_integer(Uu))),one_one(int))) ).

% ATP.lambda_66
tff(fact_7608_ATP_Olambda__67,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ik(nat,fun(nat,A)),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uua),zero_zero(nat)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),Uu))),one_one(A))) ) ).

% ATP.lambda_67
tff(fact_7609_ATP_Olambda__68,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_jx(code_integer,fun(code_integer,nat)),Uu),Uua) = if(nat,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),code_nat_of_integer(Uu)),code_nat_of_integer(Uu)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),one_one(nat))) ).

% ATP.lambda_68
tff(fact_7610_ATP_Olambda__69,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_lh(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uu),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).

% ATP.lambda_69
tff(fact_7611_ATP_Olambda__70,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa(nat,extended_enat,aTP_Lamp_aew(extended_enat,fun(nat,extended_enat),Uu),Uua) = extended_case_enat(extended_enat,aTP_Lamp_aev(nat,fun(nat,extended_enat),Uua),if(extended_enat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uua),zero_zero(nat)),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)),Uu) ).

% ATP.lambda_70
tff(fact_7612_ATP_Olambda__71,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aeg(extended_enat,fun(nat,bool),Uu),Uua))
    <=> pp(extended_case_enat(bool,aa(nat,fun(nat,bool),aTP_Lamp_ac(nat,fun(nat,bool)),Uua),fFalse,Uu)) ) ).

% ATP.lambda_71
tff(fact_7613_ATP_Olambda__72,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dw(nat,fun(nat,A),Uu),Uua) = if(A,aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_72
tff(fact_7614_ATP_Olambda__73,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_afm(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),top_top(A)) ) ).

% ATP.lambda_73
tff(fact_7615_ATP_Olambda__74,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_lx(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),bot_bot(A)) ) ).

% ATP.lambda_74
tff(fact_7616_ATP_Olambda__75,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aTP_Lamp_kb(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),Uu),Uua))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua))) ) ).

% ATP.lambda_75
tff(fact_7617_ATP_Olambda__76,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_afo(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),Uu,Uua),Uua) ) ).

% ATP.lambda_76
tff(fact_7618_ATP_Olambda__77,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_mk(list(list(A)),fun(nat,list(A)),Uu),Uua) = map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mj(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua),upt(zero_zero(nat),aa(list(list(A)),nat,size_size(list(list(A))),Uu))) ).

% ATP.lambda_77
tff(fact_7619_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_if(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ie(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_78
tff(fact_7620_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_id(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ic(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_79
tff(fact_7621_ATP_Olambda__80,axiom,
    ! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_aed(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uu)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).

% ATP.lambda_80
tff(fact_7622_ATP_Olambda__81,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_hb(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uu),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_81
tff(fact_7623_ATP_Olambda__82,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_fz(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(nat,real,aa(real,fun(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,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_82
tff(fact_7624_ATP_Olambda__83,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_gf(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(nat,real,aa(real,fun(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,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_83
tff(fact_7625_ATP_Olambda__84,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fv(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_84
tff(fact_7626_ATP_Olambda__85,axiom,
    ! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_aec(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).

% ATP.lambda_85
tff(fact_7627_ATP_Olambda__86,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_gh(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))) ).

% ATP.lambda_86
tff(fact_7628_ATP_Olambda__87,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_oe(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_87
tff(fact_7629_ATP_Olambda__88,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ti(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_88
tff(fact_7630_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fp(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_89
tff(fact_7631_ATP_Olambda__90,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dz(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Uu,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_90
tff(fact_7632_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fr(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).

% ATP.lambda_91
tff(fact_7633_ATP_Olambda__92,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aax(nat,fun(nat,bool)),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_92
tff(fact_7634_ATP_Olambda__93,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aTP_Lamp_xl(set(set(A)),fun(set(set(A)),bool),Uu),Uua))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Uua),Uu))
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_93
tff(fact_7635_ATP_Olambda__94,axiom,
    ! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_lv(set(option(A)),fun(option(A),bool),Uu),Uua))
    <=> ( pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),Uua),Uu))
        & ( Uua != none(A) ) ) ) ).

% ATP.lambda_94
tff(fact_7636_ATP_Olambda__95,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fb(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% ATP.lambda_95
tff(fact_7637_ATP_Olambda__96,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),aTP_Lamp_ace(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),Uu),Uua))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),Uu),Uua))
        & ! [A7: A,B5: A,C4: A] :
            ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B5)),Uua))
              & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),C4)),Uu)) )
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B5)),Uu)) ) ) ) ).

% ATP.lambda_96
tff(fact_7638_ATP_Olambda__97,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_he(real,fun(nat,real),Uu),Uua) = divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_97
tff(fact_7639_ATP_Olambda__98,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mi(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uua)),Uua) ).

% ATP.lambda_98
tff(fact_7640_ATP_Olambda__99,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_en(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uua) ).

% ATP.lambda_99
tff(fact_7641_ATP_Olambda__100,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_zv(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),insert(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_100
tff(fact_7642_ATP_Olambda__101,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_adk(A,fun(B,set(product_prod(A,B))),Uu),Uua) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).

% ATP.lambda_101
tff(fact_7643_ATP_Olambda__102,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa(complex,bool,aTP_Lamp_an(nat,fun(complex,bool),Uu),Uua))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_102
tff(fact_7644_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_am(nat,fun(A,bool),Uu),Uua))
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_103
tff(fact_7645_ATP_Olambda__104,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_it(A,fun(A,bool),Uu),Uua))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu)) ) ) ) ).

% ATP.lambda_104
tff(fact_7646_ATP_Olambda__105,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aTP_Lamp_ajw(set(B),fun(fun(A,B),bool),Uu),Uua))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image2(A,B,Uua,top_top(set(A)))),Uu)) ) ).

% ATP.lambda_105
tff(fact_7647_ATP_Olambda__106,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_ve(real,fun(real,real),Uu),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),Uu),Uua)),divide_divide(real,one_one(real),Uua)) ).

% ATP.lambda_106
tff(fact_7648_ATP_Olambda__107,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_gg(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_107
tff(fact_7649_ATP_Olambda__108,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aal(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ) ).

% ATP.lambda_108
tff(fact_7650_ATP_Olambda__109,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aaq(nat,fun(nat,bool),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu))) ) ) ).

% ATP.lambda_109
tff(fact_7651_ATP_Olambda__110,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dp(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_110
tff(fact_7652_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bs(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_111
tff(fact_7653_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ts(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_112
tff(fact_7654_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_113
tff(fact_7655_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_je(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_114
tff(fact_7656_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_jc(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_115
tff(fact_7657_ATP_Olambda__116,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ay(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_116
tff(fact_7658_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gl(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_117
tff(fact_7659_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_az(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_118
tff(fact_7660_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gu(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_119
tff(fact_7661_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_br(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_120
tff(fact_7662_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_121
tff(fact_7663_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_tr(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_122
tff(fact_7664_ATP_Olambda__123,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dg(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_123
tff(fact_7665_ATP_Olambda__124,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_wd(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y5: A] :
                ( pp(aa(A,bool,Uu,Y5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y5)) ) ) ) ) ).

% ATP.lambda_124
tff(fact_7666_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_wf(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y5: A] :
                ( pp(aa(A,bool,Uu,Y5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y5),Uua)) ) ) ) ) ).

% ATP.lambda_125
tff(fact_7667_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ek(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_126
tff(fact_7668_ATP_Olambda__127,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_eh(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_127
tff(fact_7669_ATP_Olambda__128,axiom,
    ! [B: $tType,Uu: fun(nat,sum_sum(B,nat)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ags(fun(nat,sum_sum(B,nat)),fun(nat,bool),Uu),Uua))
    <=> ( aa(nat,sum_sum(B,nat),Uu,Uua) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ) ) ).

% ATP.lambda_128
tff(fact_7670_ATP_Olambda__129,axiom,
    ! [A: $tType,Uu: fun(nat,sum_sum(A,nat)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ago(fun(nat,sum_sum(A,nat)),fun(nat,bool),Uu),Uua))
    <=> ( aa(nat,sum_sum(A,nat),Uu,Uua) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ) ) ).

% ATP.lambda_129
tff(fact_7671_ATP_Olambda__130,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,A,aTP_Lamp_abs(fun(A,real),fun(A,A),Uu),Uua) = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(A,real,Uu,Uua)),Uua) ) ).

% ATP.lambda_130
tff(fact_7672_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_afp(fun(A,A),fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,Uu,Uua)),Uua)) ) ) ).

% ATP.lambda_131
tff(fact_7673_ATP_Olambda__132,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,product_prod(nat,A),aTP_Lamp_afb(fun(A,nat),fun(A,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(A,nat,Uu,Uua)),Uua) ).

% ATP.lambda_132
tff(fact_7674_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rh(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_133
tff(fact_7675_ATP_Olambda__134,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_bj(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = zero_zero(A) ) ) ) ).

% ATP.lambda_134
tff(fact_7676_ATP_Olambda__135,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: fun(A,B),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aim(fun(A,B),fun(A,bool),Uu),Uua))
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_135
tff(fact_7677_ATP_Olambda__136,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ul(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ti(fun(nat,real),fun(nat,real),Uu)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_136
tff(fact_7678_ATP_Olambda__137,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_uk(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ti(fun(nat,real),fun(nat,real),Uu)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_137
tff(fact_7679_ATP_Olambda__138,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_mp(list(A),fun(list(A),list(list(A))),Uu),Uua) = map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_mo(list(A),fun(A,list(A))),Uua),Uu) ).

% ATP.lambda_138
tff(fact_7680_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ih(nat,fun(nat,nat)),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uu) ).

% ATP.lambda_139
tff(fact_7681_ATP_Olambda__140,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_us(fun(A,B),fun(A,real),Uu),Uua) = divide_divide(real,real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)),real_V7770717601297561774m_norm(A,Uua)) ) ).

% ATP.lambda_140
tff(fact_7682_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hn(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,suc,Uua)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_141
tff(fact_7683_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_afr(set(A),fun(set(A),bool),Uu),Uua))
        <=> ( ~ real_V358717886546972837endent(A,Uua)
            & ( real_Vector_span(A,Uua) = real_Vector_span(A,Uu) ) ) ) ) ).

% ATP.lambda_142
tff(fact_7684_ATP_Olambda__143,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ho(A,fun(nat,A),Uu),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,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_143
tff(fact_7685_ATP_Olambda__144,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_at(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_144
tff(fact_7686_ATP_Olambda__145,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_nh(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Uua)),Uu)) ).

% ATP.lambda_145
tff(fact_7687_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ug(A,fun(nat,A),Uu),Uua) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_146
tff(fact_7688_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_uf(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_147
tff(fact_7689_ATP_Olambda__148,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ga(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_148
tff(fact_7690_ATP_Olambda__149,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_gc(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_149
tff(fact_7691_ATP_Olambda__150,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: option(product_prod(A,B))] :
      ( pp(aa(option(product_prod(A,B)),bool,aTP_Lamp_ahi(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),Uu),Uua))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(option(product_prod(A,B)),set(product_prod(A,B)),set_option(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).

% ATP.lambda_150
tff(fact_7692_ATP_Olambda__151,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,set(old_node(A,product_unit)),aa(nat,fun(nat,set(old_node(A,product_unit))),aTP_Lamp_ahb(nat,fun(nat,set(old_node(A,product_unit)))),Uu),Uua) = old_Scons(A,product_unit,nth_item(A,Uu),nth_item(A,Uua)) ) ).

% ATP.lambda_151
tff(fact_7693_ATP_Olambda__152,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,rat,aa(nat,fun(nat,rat),aTP_Lamp_afh(nat,fun(nat,rat)),Uu),Uua) = fract(aa(nat,int,nat_int_decode,Uu),aa(nat,int,nat_int_decode,Uua)) ).

% ATP.lambda_152
tff(fact_7694_ATP_Olambda__153,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_kf(list(A),fun(list(A),bool)),Uu),Uua))
    <=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_153
tff(fact_7695_ATP_Olambda__154,axiom,
    ! [A: $tType,Uu: set(A),Uua: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_ahu(set(A),fun(option(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(option(A),set(A),set_option(A),Uua)),Uu)) ) ).

% ATP.lambda_154
tff(fact_7696_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fg(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_155
tff(fact_7697_ATP_Olambda__156,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_mn(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).

% ATP.lambda_156
tff(fact_7698_ATP_Olambda__157,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_nz(set(nat),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)),Uu)) ) ).

% ATP.lambda_157
tff(fact_7699_ATP_Olambda__158,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ks(set(nat),fun(nat,bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,Uua)),Uu)) ) ).

% ATP.lambda_158
tff(fact_7700_ATP_Olambda__159,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_ht(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_159
tff(fact_7701_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_hu(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_160
tff(fact_7702_ATP_Olambda__161,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cs(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% ATP.lambda_161
tff(fact_7703_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fj(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu),Uua)) ).

% ATP.lambda_162
tff(fact_7704_ATP_Olambda__163,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ud(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),Uu),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_163
tff(fact_7705_ATP_Olambda__164,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ua(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or1337092689740270186AtMost(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_164
tff(fact_7706_ATP_Olambda__165,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_afy(fun(A,A),fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),aa(A,A,Uu,Uua))) ) ) ).

% ATP.lambda_165
tff(fact_7707_ATP_Olambda__166,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_md(fun(nat,A),fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),aa(nat,A,Uu,Uua)) ).

% ATP.lambda_166
tff(fact_7708_ATP_Olambda__167,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_xh(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu,Uua)) ).

% ATP.lambda_167
tff(fact_7709_ATP_Olambda__168,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_aeu(nat,fun(vEBT_VEBT,vEBT_VEBT),Uu),Uua) = vEBT_VEBT_elim_dead(Uua,extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Uu,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% ATP.lambda_168
tff(fact_7710_ATP_Olambda__169,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_jw(A,fun(nat,A),Uu),Uua) = aa(A,A,bit_se4730199178511100633sh_bit(A,Uua),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_169
tff(fact_7711_ATP_Olambda__170,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_abb(fun(A,option(B)),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(option(B),B,the2(B),aa(A,option(B),Uu,Uua))) ).

% ATP.lambda_170
tff(fact_7712_ATP_Olambda__171,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_ii(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),zero_zero(int))))) ).

% ATP.lambda_171
tff(fact_7713_ATP_Olambda__172,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ub(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu),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_172
tff(fact_7714_ATP_Olambda__173,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_tw(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))) ).

% ATP.lambda_173
tff(fact_7715_ATP_Olambda__174,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_aap(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ).

% ATP.lambda_174
tff(fact_7716_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_to(A,fun(nat,A),Uu),Uua) = divide_divide(A,Uu,aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_175
tff(fact_7717_ATP_Olambda__176,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_nj(nat,fun(list(A),bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua))) ) ).

% ATP.lambda_176
tff(fact_7718_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: A,Uua: real] : aa(real,A,aTP_Lamp_afu(A,fun(real,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),real_Vector_of_real(A,Uua)) ) ).

% ATP.lambda_177
tff(fact_7719_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fs(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_178
tff(fact_7720_ATP_Olambda__179,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_jl(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_179
tff(fact_7721_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dk(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_180
tff(fact_7722_ATP_Olambda__181,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,list(nat),aa(nat,fun(nat,list(nat)),aTP_Lamp_jq(nat,fun(nat,list(nat))),Uu),Uua) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Uu),aa(nat,list(nat),nat_list_decode,Uua)) ).

% ATP.lambda_181
tff(fact_7723_ATP_Olambda__182,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_zu(nat,fun(nat,bool)),Uu),Uua))
    <=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).

% ATP.lambda_182
tff(fact_7724_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ac(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uu)) ) ).

% ATP.lambda_183
tff(fact_7725_ATP_Olambda__184,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_kh(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_184
tff(fact_7726_ATP_Olambda__185,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aeo(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_185
tff(fact_7727_ATP_Olambda__186,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ady(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_186
tff(fact_7728_ATP_Olambda__187,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aeh(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_187
tff(fact_7729_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_lw(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_188
tff(fact_7730_ATP_Olambda__189,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aio(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_189
tff(fact_7731_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ec(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_190
tff(fact_7732_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_nd(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_191
tff(fact_7733_ATP_Olambda__192,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_kw(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_192
tff(fact_7734_ATP_Olambda__193,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_wu(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_193
tff(fact_7735_ATP_Olambda__194,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_bt(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).

% ATP.lambda_194
tff(fact_7736_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ki(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_195
tff(fact_7737_ATP_Olambda__196,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aep(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_196
tff(fact_7738_ATP_Olambda__197,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_adz(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_197
tff(fact_7739_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aig(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_198
tff(fact_7740_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aen(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_199
tff(fact_7741_ATP_Olambda__200,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aip(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_200
tff(fact_7742_ATP_Olambda__201,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_cz(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_201
tff(fact_7743_ATP_Olambda__202,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_tn(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_202
tff(fact_7744_ATP_Olambda__203,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_uo(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_203
tff(fact_7745_ATP_Olambda__204,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_pg(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_204
tff(fact_7746_ATP_Olambda__205,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_kz(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_205
tff(fact_7747_ATP_Olambda__206,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_kt(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_206
tff(fact_7748_ATP_Olambda__207,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_kv(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_207
tff(fact_7749_ATP_Olambda__208,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_pf(nat,fun(real,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu) ).

% ATP.lambda_208
tff(fact_7750_ATP_Olambda__209,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_xp(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).

% ATP.lambda_209
tff(fact_7751_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ww(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ).

% ATP.lambda_210
tff(fact_7752_ATP_Olambda__211,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mr(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_211
tff(fact_7753_ATP_Olambda__212,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_lm(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_212
tff(fact_7754_ATP_Olambda__213,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_acd(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_213
tff(fact_7755_ATP_Olambda__214,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ku(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_214
tff(fact_7756_ATP_Olambda__215,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_om(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_215
tff(fact_7757_ATP_Olambda__216,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_bo(nat,fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uua),Uu)) ) ).

% ATP.lambda_216
tff(fact_7758_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_bh(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),Uu)) ) ) ).

% ATP.lambda_217
tff(fact_7759_ATP_Olambda__218,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: set(A)] : aa(set(A),set(product_prod(A,B)),aTP_Lamp_act(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),Uu),Uua) = product_Sigma(A,B,Uua,Uu) ).

% ATP.lambda_218
tff(fact_7760_ATP_Olambda__219,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_kl(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uu) ).

% ATP.lambda_219
tff(fact_7761_ATP_Olambda__220,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_mv(B,fun(A,product_prod(A,B))),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).

% ATP.lambda_220
tff(fact_7762_ATP_Olambda__221,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_my(A,fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),Uu) ).

% ATP.lambda_221
tff(fact_7763_ATP_Olambda__222,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_xi(A,fun(B,product_prod(B,A))),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu) ).

% ATP.lambda_222
tff(fact_7764_ATP_Olambda__223,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ed(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_223
tff(fact_7765_ATP_Olambda__224,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_mo(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uu) ).

% ATP.lambda_224
tff(fact_7766_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: int,Uua: A] : aa(A,A,aTP_Lamp_zh(int,fun(A,A),Uu),Uua) = power_int(A,Uua,Uu) ) ).

% ATP.lambda_225
tff(fact_7767_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_iz(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ) ).

% ATP.lambda_226
tff(fact_7768_ATP_Olambda__227,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_mf(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ) ).

% ATP.lambda_227
tff(fact_7769_ATP_Olambda__228,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_a(set(A),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ).

% ATP.lambda_228
tff(fact_7770_ATP_Olambda__229,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_aak(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),Uu),Uua) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Uua),Uu) ).

% ATP.lambda_229
tff(fact_7771_ATP_Olambda__230,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_nk(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_230
tff(fact_7772_ATP_Olambda__231,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_afs(A,fun(A,bool),Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_231
tff(fact_7773_ATP_Olambda__232,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_adt(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_232
tff(fact_7774_ATP_Olambda__233,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_add(set(product_prod(A,B)),fun(A,set(B)),Uu),Uua) = image2(product_prod(A,B),B,product_snd(A,B),Uu) ).

% ATP.lambda_233
tff(fact_7775_ATP_Olambda__234,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_dv(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))) ).

% ATP.lambda_234
tff(fact_7776_ATP_Olambda__235,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_du(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)) ).

% ATP.lambda_235
tff(fact_7777_ATP_Olambda__236,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_uv(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,Uu,divide_divide(A,one_one(A),Uua)) ) ).

% ATP.lambda_236
tff(fact_7778_ATP_Olambda__237,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_nw(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)))) ) ).

% ATP.lambda_237
tff(fact_7779_ATP_Olambda__238,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_nx(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).

% ATP.lambda_238
tff(fact_7780_ATP_Olambda__239,axiom,
    ! [Uu: fun(nat,bool),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_iy(fun(nat,bool),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_239
tff(fact_7781_ATP_Olambda__240,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_240
tff(fact_7782_ATP_Olambda__241,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_tm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_241
tff(fact_7783_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cr(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_242
tff(fact_7784_ATP_Olambda__243,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_da(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_243
tff(fact_7785_ATP_Olambda__244,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bk(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_244
tff(fact_7786_ATP_Olambda__245,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_mh(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_245
tff(fact_7787_ATP_Olambda__246,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_zx(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),ord_min(A),Uu)),Uua)) ) ).

% ATP.lambda_246
tff(fact_7788_ATP_Olambda__247,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_aab(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),ord_max(A),Uu)),Uua)) ) ).

% ATP.lambda_247
tff(fact_7789_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_aah(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),sup_sup(A),Uu)),Uua)) ) ).

% ATP.lambda_248
tff(fact_7790_ATP_Olambda__249,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_aai(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),inf_inf(A),Uu)),Uua)) ) ).

% ATP.lambda_249
tff(fact_7791_ATP_Olambda__250,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_kq(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_250
tff(fact_7792_ATP_Olambda__251,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_ko(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_251
tff(fact_7793_ATP_Olambda__252,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_kn(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_km(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_252
tff(fact_7794_ATP_Olambda__253,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kk(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_kj(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_253
tff(fact_7795_ATP_Olambda__254,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_ov(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_ou(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_254
tff(fact_7796_ATP_Olambda__255,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_ok(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_255
tff(fact_7797_ATP_Olambda__256,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_xy(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_xv(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),top_top(set(B)))) ) ).

% ATP.lambda_256
tff(fact_7798_ATP_Olambda__257,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_xu(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_xt(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),top_top(set(C)))) ) ).

% ATP.lambda_257
tff(fact_7799_ATP_Olambda__258,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_xw(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_xv(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),top_top(set(B)))) ) ).

% ATP.lambda_258
tff(fact_7800_ATP_Olambda__259,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_xx(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),image2(C,A,aa(B,fun(C,A),aTP_Lamp_xt(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),top_top(set(C)))) ) ).

% ATP.lambda_259
tff(fact_7801_ATP_Olambda__260,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_iq(nat,fun(nat,complex),Uu),Uua) = cis(divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(nat,real,semiring_1_of_nat(real),Uua)),aa(nat,real,semiring_1_of_nat(real),Uu))) ).

% ATP.lambda_260
tff(fact_7802_ATP_Olambda__261,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_qj(A,fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Uu)),Uua)),aa(A,A,inverse_inverse(A),Uu))) ) ).

% ATP.lambda_261
tff(fact_7803_ATP_Olambda__262,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_abt(fun(A,real),fun(A,bool),Uu),Uua))
        <=> ( aa(A,real,Uu,Uua) != zero_zero(real) ) ) ) ).

% ATP.lambda_262
tff(fact_7804_ATP_Olambda__263,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_adl(fun(A,set(B)),fun(A,set(product_prod(A,B))),Uu),Uua) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image2(B,set(product_prod(A,B)),aTP_Lamp_adk(A,fun(B,set(product_prod(A,B))),Uua),aa(A,set(B),Uu,Uua))) ).

% ATP.lambda_263
tff(fact_7805_ATP_Olambda__264,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: list(product_prod(C,B)),Uua: product_prod(A,C)] : aa(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_zg(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = concat(product_prod(A,B),map(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_zf(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua),Uu)) ).

% ATP.lambda_264
tff(fact_7806_ATP_Olambda__265,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_fw(nat,fun(nat,bool),Uu),Uua))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Uu,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_265
tff(fact_7807_ATP_Olambda__266,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_aih(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Uua),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),Uu),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))))))))))))))))))))) ).

% ATP.lambda_266
tff(fact_7808_ATP_Olambda__267,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_xa(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image2(nat,set(A),Uu,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_267
tff(fact_7809_ATP_Olambda__268,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fo(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_268
tff(fact_7810_ATP_Olambda__269,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_cv(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_269
tff(fact_7811_ATP_Olambda__270,axiom,
    ! [A: $tType,Uu: list(A),Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),aTP_Lamp_aik(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(A,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),product_Pair(A,product_prod(code_natural,code_natural)),aa(nat,A,nth(A,Uu),aa(code_natural,nat,code_nat_of_natural,Uua))) ).

% ATP.lambda_270
tff(fact_7812_ATP_Olambda__271,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_yq(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),image2(B,A,Uu,Uua)) ) ).

% ATP.lambda_271
tff(fact_7813_ATP_Olambda__272,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_yp(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),image2(B,A,Uu,Uua)) ) ).

% ATP.lambda_272
tff(fact_7814_ATP_Olambda__273,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_adj(nat,fun(nat,set(nat)),Uu),Uua) = aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uua)) ).

% ATP.lambda_273
tff(fact_7815_ATP_Olambda__274,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_aij(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),modulo_modulo(code_natural,Uua,Uu)) ).

% ATP.lambda_274
tff(fact_7816_ATP_Olambda__275,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(C,product_prod(product_prod(A,B),C)),aTP_Lamp_aca(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu),Uua) = aa(product_prod(A,B),fun(C,product_prod(product_prod(A,B),C)),product_Pair(product_prod(A,B),C),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_275
tff(fact_7817_ATP_Olambda__276,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,extended_enat,aTP_Lamp_aev(nat,fun(nat,extended_enat),Uu),Uua) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uua)) ).

% ATP.lambda_276
tff(fact_7818_ATP_Olambda__277,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_zw(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = insert(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).

% ATP.lambda_277
tff(fact_7819_ATP_Olambda__278,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_adm(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua) = insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_278
tff(fact_7820_ATP_Olambda__279,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zd(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uu),Uua)) ).

% ATP.lambda_279
tff(fact_7821_ATP_Olambda__280,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ze(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uua),Uu)) ).

% ATP.lambda_280
tff(fact_7822_ATP_Olambda__281,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jv(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uu),Uua)) ).

% ATP.lambda_281
tff(fact_7823_ATP_Olambda__282,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ju(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uua),Uu)) ).

% ATP.lambda_282
tff(fact_7824_ATP_Olambda__283,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_afn(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_lfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_283
tff(fact_7825_ATP_Olambda__284,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_afw(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_gfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_284
tff(fact_7826_ATP_Olambda__285,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_gx(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_285
tff(fact_7827_ATP_Olambda__286,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_gr(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_286
tff(fact_7828_ATP_Olambda__287,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_se(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_287
tff(fact_7829_ATP_Olambda__288,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_tu(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_288
tff(fact_7830_ATP_Olambda__289,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_qk(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_289
tff(fact_7831_ATP_Olambda__290,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_sk(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_290
tff(fact_7832_ATP_Olambda__291,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_vp(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_291
tff(fact_7833_ATP_Olambda__292,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wq(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_292
tff(fact_7834_ATP_Olambda__293,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pd(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_293
tff(fact_7835_ATP_Olambda__294,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_sx(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_294
tff(fact_7836_ATP_Olambda__295,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_vc(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_295
tff(fact_7837_ATP_Olambda__296,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qs(fun(A,real),fun(A,real),Uu),Uua) = ln_ln(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_296
tff(fact_7838_ATP_Olambda__297,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_db(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_297
tff(fact_7839_ATP_Olambda__298,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_dq(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_298
tff(fact_7840_ATP_Olambda__299,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pt(fun(A,real),fun(A,real),Uu),Uua) = arctan(aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_299
tff(fact_7841_ATP_Olambda__300,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_re(fun(A,real),fun(A,real),Uu),Uua) = arcsin(aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_300
tff(fact_7842_ATP_Olambda__301,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pp(fun(A,real),fun(A,real),Uu),Uua) = arccos(aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_301
tff(fact_7843_ATP_Olambda__302,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_sh(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,sgn_sgn(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_302
tff(fact_7844_ATP_Olambda__303,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wr(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_303
tff(fact_7845_ATP_Olambda__304,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_sy(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_304
tff(fact_7846_ATP_Olambda__305,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_gy(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_305
tff(fact_7847_ATP_Olambda__306,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_gd(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,abs_abs(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_306
tff(fact_7848_ATP_Olambda__307,axiom,
    ! [A9: $tType] :
      ( ( real_Vector_banach(A9)
        & real_V3459762299906320749_field(A9) )
     => ! [Uu: fun(A9,A9),Uua: A9] : aa(A9,A9,aTP_Lamp_os(fun(A9,A9),fun(A9,A9),Uu),Uua) = aa(A9,A9,tanh(A9),aa(A9,A9,Uu,Uua)) ) ).

% ATP.lambda_307
tff(fact_7849_ATP_Olambda__308,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_wt(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_308
tff(fact_7850_ATP_Olambda__309,axiom,
    ! [A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_tc(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_309
tff(fact_7851_ATP_Olambda__310,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: C] : aa(C,A,aTP_Lamp_sm(fun(C,A),fun(C,A),Uu),Uua) = aa(A,A,tanh(A),aa(C,A,Uu,Uua)) ) ).

% ATP.lambda_310
tff(fact_7852_ATP_Olambda__311,axiom,
    ! [A9: $tType] :
      ( ( real_Vector_banach(A9)
        & real_V3459762299906320749_field(A9) )
     => ! [Uu: fun(A9,A9),Uua: A9] : aa(A9,A9,aTP_Lamp_of(fun(A9,A9),fun(A9,A9),Uu),Uua) = sinh(A9,aa(A9,A9,Uu,Uua)) ) ).

% ATP.lambda_311
tff(fact_7853_ATP_Olambda__312,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_qf(fun(A,A),fun(A,A),Uu),Uua) = sinh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_312
tff(fact_7854_ATP_Olambda__313,axiom,
    ! [A9: $tType] :
      ( ( real_Vector_banach(A9)
        & real_V3459762299906320749_field(A9) )
     => ! [Uu: fun(A9,A9),Uua: A9] : aa(A9,A9,aTP_Lamp_og(fun(A9,A9),fun(A9,A9),Uu),Uua) = cosh(A9,aa(A9,A9,Uu,Uua)) ) ).

% ATP.lambda_313
tff(fact_7855_ATP_Olambda__314,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_qg(fun(A,A),fun(A,A),Uu),Uua) = cosh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_314
tff(fact_7856_ATP_Olambda__315,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pr(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,tan(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_315
tff(fact_7857_ATP_Olambda__316,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rn(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_316
tff(fact_7858_ATP_Olambda__317,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qd(fun(A,real),fun(A,real),Uu),Uua) = sin(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_317
tff(fact_7859_ATP_Olambda__318,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_oi(fun(A,A),fun(A,A),Uu),Uua) = sin(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_318
tff(fact_7860_ATP_Olambda__319,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qb(fun(A,real),fun(A,real),Uu),Uua) = exp(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_319
tff(fact_7861_ATP_Olambda__320,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_oh(fun(A,A),fun(A,A),Uu),Uua) = exp(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_320
tff(fact_7862_ATP_Olambda__321,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_sn(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_321
tff(fact_7863_ATP_Olambda__322,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_uj(fun(nat,real),fun(nat,real),Uu),Uua) = cos(real,aa(nat,real,Uu,Uua)) ).

% ATP.lambda_322
tff(fact_7864_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qo(fun(A,real),fun(A,real),Uu),Uua) = cos(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_323
tff(fact_7865_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_oj(fun(A,A),fun(A,A),Uu),Uua) = cos(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_324
tff(fact_7866_ATP_Olambda__325,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_afg(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_325
tff(fact_7867_ATP_Olambda__326,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_aaz(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_326
tff(fact_7868_ATP_Olambda__327,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_ly(fun(C,A),fun(C,fun(B,product_prod(A,B))),Uu),Uua) = aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uua)) ).

% ATP.lambda_327
tff(fact_7869_ATP_Olambda__328,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_xd(fun(A,B),fun(A,fun(C,product_prod(B,C))),Uu),Uua) = aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uua)) ).

% ATP.lambda_328
tff(fact_7870_ATP_Olambda__329,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_xc(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_329
tff(fact_7871_ATP_Olambda__330,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_oo(fun(real,fun(nat,real)),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),Uu,Uua)) ).

% ATP.lambda_330
tff(fact_7872_ATP_Olambda__331,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_td(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_331
tff(fact_7873_ATP_Olambda__332,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rc(fun(A,real),fun(A,real),Uu),Uua) = sqrt(aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_332
tff(fact_7874_ATP_Olambda__333,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: A] : aa(A,set(B),aTP_Lamp_acp(fun(A,fun(B,bool)),fun(A,set(B)),Uu),Uua) = aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),Uu,Uua)) ).

% ATP.lambda_333
tff(fact_7875_ATP_Olambda__334,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_cf(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_334
tff(fact_7876_ATP_Olambda__335,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_nl(fun(A,bool),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(A,bool,Uu,Uua)) ) ).

% ATP.lambda_335
tff(fact_7877_ATP_Olambda__336,axiom,
    ! [A: $tType,B: $tType,Uu: fun(nat,sum_sum(B,nat)),Uua: sum_sum(A,nat)] : aa(sum_sum(A,nat),nat,aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat),aTP_Lamp_agt(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat)),Uu),Uua) = ord_Least(nat,aTP_Lamp_ags(fun(nat,sum_sum(B,nat)),fun(nat,bool),Uu)) ).

% ATP.lambda_336
tff(fact_7878_ATP_Olambda__337,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_adr(nat,fun(nat,set(nat)),Uu),Uua) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_bt(nat,fun(nat,bool)),Uu)) ).

% ATP.lambda_337
tff(fact_7879_ATP_Olambda__338,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_yk(list(A),fun(A,bool),Uu),Uua))
    <=> ? [I: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Uu))) ) ) ).

% ATP.lambda_338
tff(fact_7880_ATP_Olambda__339,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: set(B)] :
      ( pp(aa(set(B),bool,aTP_Lamp_yr(set(set(B)),fun(set(B),bool),Uu),Uua))
    <=> ? [F5: fun(set(B),B)] :
          ( ( Uua = image2(set(B),B,F5,Uu) )
          & ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X4),Uu))
             => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(set(B),B,F5,X4)),X4)) ) ) ) ).

% ATP.lambda_339
tff(fact_7881_ATP_Olambda__340,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_yn(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = image2(set(A),A,F5,Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F5,X4)),X4)) ) ) ) ) ).

% ATP.lambda_340
tff(fact_7882_ATP_Olambda__341,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_yo(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = image2(set(A),A,F5,Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F5,X4)),X4)) ) ) ) ) ).

% ATP.lambda_341
tff(fact_7883_ATP_Olambda__342,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_ys(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = image2(set(A),A,F5,Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F5,X4)),X4)) ) ) ) ) ).

% ATP.lambda_342
tff(fact_7884_ATP_Olambda__343,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aeq(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X4)) ) ) ) ).

% ATP.lambda_343
tff(fact_7885_ATP_Olambda__344,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_xr(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X4)) ) ) ) ).

% ATP.lambda_344
tff(fact_7886_ATP_Olambda__345,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aem(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Uua)) ) ) ) ).

% ATP.lambda_345
tff(fact_7887_ATP_Olambda__346,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_xs(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Uua)) ) ) ) ).

% ATP.lambda_346
tff(fact_7888_ATP_Olambda__347,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_wc(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y5))
             => pp(aa(A,bool,Uu,Y5)) ) ) ) ).

% ATP.lambda_347
tff(fact_7889_ATP_Olambda__348,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_agd(set(product_prod(A,A)),fun(set(A),bool),Uu),Uua))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uua))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),Uua))
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),Uu))
                | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X4)),Uu)) ) ) ) ) ).

% ATP.lambda_348
tff(fact_7890_ATP_Olambda__349,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_agp(set(product_prod(A,B)),fun(A,bool),Uu),Uua))
    <=> ? [Y5: B] : pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Y5)),Uu)) ) ).

% ATP.lambda_349
tff(fact_7891_ATP_Olambda__350,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_zo(fun(A,option(B)),fun(B,bool),Uu),Uua))
    <=> ? [A7: A] : aa(A,option(B),Uu,A7) = aa(B,option(B),some(B),Uua) ) ).

% ATP.lambda_350
tff(fact_7892_ATP_Olambda__351,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aTP_Lamp_abf(fun(A,option(B)),fun(product_prod(A,B),bool),Uu),Uua))
    <=> ? [A7: A,B5: B] :
          ( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5) )
          & ( aa(A,option(B),Uu,A7) = aa(B,option(B),some(B),B5) ) ) ) ).

% ATP.lambda_351
tff(fact_7893_ATP_Olambda__352,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
      ( pp(aa(product_prod(set(A),set(A)),bool,aTP_Lamp_agc(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),Uu),Uua))
    <=> ? [X7: set(A),Y9: set(A)] :
          ( ( Uua = aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X7),Y9) )
          & ( X7 != bot_bot(set(A)) )
          & ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Y9))
             => ? [Xa3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),X7))
                  & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X4)),Uu)) ) ) ) ) ).

% ATP.lambda_352
tff(fact_7894_ATP_Olambda__353,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_acj(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).

% ATP.lambda_353
tff(fact_7895_ATP_Olambda__354,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_aea(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).

% ATP.lambda_354
tff(fact_7896_ATP_Olambda__355,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: C] : aa(C,set(B),aTP_Lamp_ajr(set(product_prod(B,B)),fun(C,set(B)),Uu),Uua) = field2(B,Uu) ).

% ATP.lambda_355
tff(fact_7897_ATP_Olambda__356,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: A] : aa(A,set(B),aTP_Lamp_ajt(set(product_prod(B,B)),fun(A,set(B)),Uu),Uua) = field2(B,Uu) ).

% ATP.lambda_356
tff(fact_7898_ATP_Olambda__357,axiom,
    ! [C: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: C] : aa(C,set(A),aTP_Lamp_ajq(set(product_prod(A,A)),fun(C,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_357
tff(fact_7899_ATP_Olambda__358,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: B] : aa(B,set(A),aTP_Lamp_ajn(set(product_prod(A,A)),fun(B,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_358
tff(fact_7900_ATP_Olambda__359,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_ajm(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_359
tff(fact_7901_ATP_Olambda__360,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: A] : aa(A,set(B),aTP_Lamp_ach(fun(B,bool),fun(A,set(B)),Uu),Uua) = aa(fun(B,bool),set(B),collect(B),Uu) ).

% ATP.lambda_360
tff(fact_7902_ATP_Olambda__361,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: option(A),Uub: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aTP_Lamp_aht(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),Uu),Uua),Uub))
    <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),aa(option(B),bool,aa(fun(B,bool),fun(option(B),bool),aa(bool,fun(fun(B,bool),fun(option(B),bool)),case_option(bool,B),fTrue),aTP_Lamp_ahr(B,bool)),Uub)),aa(option(B),fun(A,bool),aTP_Lamp_ahs(fun(A,fun(B,bool)),fun(option(B),fun(A,bool)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_361
tff(fact_7903_ATP_Olambda__362,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_dt(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = if(real,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uu,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_362
tff(fact_7904_ATP_Olambda__363,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_jp(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_363
tff(fact_7905_ATP_Olambda__364,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_hx(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = if(product_prod(nat,nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_364
tff(fact_7906_ATP_Olambda__365,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hy(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_365
tff(fact_7907_ATP_Olambda__366,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: num,Uua: A,Uub: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_hz(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = if(product_prod(A,A),aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_366
tff(fact_7908_ATP_Olambda__367,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: set(nat),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cb(set(nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = if(A,aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_367
tff(fact_7909_ATP_Olambda__368,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_bi(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uua),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_368
tff(fact_7910_ATP_Olambda__369,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aw(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),aa(B,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_369
tff(fact_7911_ATP_Olambda__370,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cj(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uub),Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_370
tff(fact_7912_ATP_Olambda__371,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_av(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),aa(B,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_371
tff(fact_7913_ATP_Olambda__372,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_aar(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),Uu)),Uub))) ).

% ATP.lambda_372
tff(fact_7914_ATP_Olambda__373,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,bool),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cc(fun(nat,bool),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = if(A,aa(nat,bool,Uu,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_373
tff(fact_7915_ATP_Olambda__374,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_bg(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_374
tff(fact_7916_ATP_Olambda__375,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_no(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_375
tff(fact_7917_ATP_Olambda__376,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,option(A),aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_ns(fun(B,A),fun(fun(B,bool),fun(B,option(A))),Uu),Uua),Uub) = if(option(A),aa(B,bool,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu,Uub)),none(A)) ).

% ATP.lambda_376
tff(fact_7918_ATP_Olambda__377,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A,Uub: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(A,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_adn(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua),Uub) = finite_fold(B,set(product_prod(A,B)),aTP_Lamp_adm(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).

% ATP.lambda_377
tff(fact_7919_ATP_Olambda__378,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B,Uub: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(B,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_aaj(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua),Uub) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_zw(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).

% ATP.lambda_378
tff(fact_7920_ATP_Olambda__379,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: option(A),Uub: A] : aa(A,option(A),aa(option(A),fun(A,option(A)),aTP_Lamp_ahk(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = aa(option(A),option(A),aa(fun(A,option(A)),fun(option(A),option(A)),aa(option(A),fun(fun(A,option(A)),fun(option(A),option(A))),case_option(option(A),A),aa(A,option(A),some(A),Uub)),aa(A,fun(A,option(A)),aTP_Lamp_ahj(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uub)),Uua) ).

% ATP.lambda_379
tff(fact_7921_ATP_Olambda__380,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] : aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_abg(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub) = fun_upd(A,option(B),Uu,Uua,aa(B,option(B),some(B),Uub)) ).

% ATP.lambda_380
tff(fact_7922_ATP_Olambda__381,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aa(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_agi(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),Uu),Uua),Uub) = fun_upd(A,option(B),Uub,Uu,aa(B,option(B),some(B),Uua)) ).

% ATP.lambda_381
tff(fact_7923_ATP_Olambda__382,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_xj(fun(A,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uub),Uu),Uua) ).

% ATP.lambda_382
tff(fact_7924_ATP_Olambda__383,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: option(B),Uub: A] :
      ( pp(aa(A,bool,aa(option(B),fun(A,bool),aTP_Lamp_ahs(fun(A,fun(B,bool)),fun(option(B),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(option(B),bool,aa(fun(B,bool),fun(option(B),bool),aa(bool,fun(fun(B,bool),fun(option(B),bool)),case_option(bool,B),fFalse),aa(A,fun(B,bool),Uu,Uub)),Uua)) ) ).

% ATP.lambda_383
tff(fact_7925_ATP_Olambda__384,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ajf(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),Uu,aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub))),aa(nat,A,Uua,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)))) ).

% ATP.lambda_384
tff(fact_7926_ATP_Olambda__385,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: product_prod(D,B)] : aa(product_prod(D,B),C,aa(fun(D,A),fun(product_prod(D,B),C),aTP_Lamp_jy(fun(A,fun(B,C)),fun(fun(D,A),fun(product_prod(D,B),C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,aa(product_prod(D,B),D,product_fst(D,B),Uub))),aa(product_prod(D,B),B,product_snd(D,B),Uub)) ).

% ATP.lambda_385
tff(fact_7927_ATP_Olambda__386,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_xv(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,aa(B,C,Uua,Uub)),Uub) ) ).

% ATP.lambda_386
tff(fact_7928_ATP_Olambda__387,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ie(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_387
tff(fact_7929_ATP_Olambda__388,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ic(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_388
tff(fact_7930_ATP_Olambda__389,axiom,
    ! [A: $tType,Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ajj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ).

% ATP.lambda_389
tff(fact_7931_ATP_Olambda__390,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_on(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),Uu,Uub),Uua) ).

% ATP.lambda_390
tff(fact_7932_ATP_Olambda__391,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_el(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_391
tff(fact_7933_ATP_Olambda__392,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ei(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_392
tff(fact_7934_ATP_Olambda__393,axiom,
    ! [A: $tType,Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ajb(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ).

% ATP.lambda_393
tff(fact_7935_ATP_Olambda__394,axiom,
    ! [I6: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(I6,fun(A,B)),Uua: A,Uub: I6] : aa(I6,B,aa(A,fun(I6,B),aTP_Lamp_qw(fun(I6,fun(A,B)),fun(A,fun(I6,B)),Uu),Uua),Uub) = aa(A,B,aa(I6,fun(A,B),Uu,Uub),Uua) ) ).

% ATP.lambda_394
tff(fact_7936_ATP_Olambda__395,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_xt(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).

% ATP.lambda_395
tff(fact_7937_ATP_Olambda__396,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_adw(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ).

% ATP.lambda_396
tff(fact_7938_ATP_Olambda__397,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_sf(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_397
tff(fact_7939_ATP_Olambda__398,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ain(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_398
tff(fact_7940_ATP_Olambda__399,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hq(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hp(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_399
tff(fact_7941_ATP_Olambda__400,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hl(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hk(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_400
tff(fact_7942_ATP_Olambda__401,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hj(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hi(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_401
tff(fact_7943_ATP_Olambda__402,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gt(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_gs(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_402
tff(fact_7944_ATP_Olambda__403,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ca(nat,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uub),Uu),one_one(A),zero_zero(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_403
tff(fact_7945_ATP_Olambda__404,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_em(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_el(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_404
tff(fact_7946_ATP_Olambda__405,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ej(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ei(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_405
tff(fact_7947_ATP_Olambda__406,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_te(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fk(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_406
tff(fact_7948_ATP_Olambda__407,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_lu(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_lt(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_407
tff(fact_7949_ATP_Olambda__408,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ls(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_lr(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_408
tff(fact_7950_ATP_Olambda__409,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_lq(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_lp(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_409
tff(fact_7951_ATP_Olambda__410,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_lo(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_ln(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_410
tff(fact_7952_ATP_Olambda__411,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ll(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_lk(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_411
tff(fact_7953_ATP_Olambda__412,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_lj(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_li(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_412
tff(fact_7954_ATP_Olambda__413,axiom,
    ! [A: $tType,B: $tType,I6: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I6),Uua: fun(I6,fun(A,B)),Uub: A] : aa(A,B,aa(fun(I6,fun(A,B)),fun(A,B),aTP_Lamp_qx(set(I6),fun(fun(I6,fun(A,B)),fun(A,B)),Uu),Uua),Uub) = aa(set(I6),B,aa(fun(I6,B),fun(set(I6),B),groups7121269368397514597t_prod(I6,B),aa(A,fun(I6,B),aTP_Lamp_qw(fun(I6,fun(A,B)),fun(A,fun(I6,B)),Uua),Uub)),Uu) ) ).

% ATP.lambda_413
tff(fact_7955_ATP_Olambda__414,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ow(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,Uub),zero_zero(real)),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_414
tff(fact_7956_ATP_Olambda__415,axiom,
    ! [Uu: real,Uua: fun(nat,fun(real,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_ox(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uua,Uub),zero_zero(real)),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_415
tff(fact_7957_ATP_Olambda__416,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Uu: real,Uua: fun(nat,fun(A,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_fy(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(A,real,aa(nat,fun(A,real),Uua,Uub),zero_zero(A)),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_416
tff(fact_7958_ATP_Olambda__417,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( pp(aa(A,bool,aa(nat,fun(A,bool),aTP_Lamp_er(fun(nat,A),fun(nat,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_eg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_417
tff(fact_7959_ATP_Olambda__418,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_sz(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua)),Uub) ) ).

% ATP.lambda_418
tff(fact_7960_ATP_Olambda__419,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_st(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua)),Uub) ) ).

% ATP.lambda_419
tff(fact_7961_ATP_Olambda__420,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ba(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_420
tff(fact_7962_ATP_Olambda__421,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_su(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),Uua)) ) ).

% ATP.lambda_421
tff(fact_7963_ATP_Olambda__422,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ot(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_422
tff(fact_7964_ATP_Olambda__423,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_fx(real,fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(nat,real,Uua,Uub),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_423
tff(fact_7965_ATP_Olambda__424,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_na(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),one_one(nat)) )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_424
tff(fact_7966_ATP_Olambda__425,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hd(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,Uu,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_425
tff(fact_7967_ATP_Olambda__426,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_ij(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub)))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub)),lex(A,Uu))) ) ) ) ).

% ATP.lambda_426
tff(fact_7968_ATP_Olambda__427,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_abn(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(nat,nat,suc,Uu) )
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uu) ) ) ) ).

% ATP.lambda_427
tff(fact_7969_ATP_Olambda__428,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_yj(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys2: list(A),X4: A,Y5: A,Xs5: list(A),Ys6: list(A)] :
            ( ( Uua = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs5)) )
            & ( Uub = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys6)) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5)),Uu)) ) ) ) ).

% ATP.lambda_428
tff(fact_7970_ATP_Olambda__429,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_nb(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_429
tff(fact_7971_ATP_Olambda__430,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(set(A),fun(list(A),bool),aTP_Lamp_fn(nat,fun(set(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua)) ) ) ).

% ATP.lambda_430
tff(fact_7972_ATP_Olambda__431,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_dy(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)) ) ) ).

% ATP.lambda_431
tff(fact_7973_ATP_Olambda__432,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_adh(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_432
tff(fact_7974_ATP_Olambda__433,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_io(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua))) ) ) ).

% ATP.lambda_433
tff(fact_7975_ATP_Olambda__434,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_ak(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua)) ) ) ).

% ATP.lambda_434
tff(fact_7976_ATP_Olambda__435,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_aj(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_435
tff(fact_7977_ATP_Olambda__436,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_mz(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_436
tff(fact_7978_ATP_Olambda__437,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_op(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu)),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_437
tff(fact_7979_ATP_Olambda__438,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ai(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,Uub)),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_438
tff(fact_7980_ATP_Olambda__439,axiom,
    ! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_oa(set(nat),fun(nat,fun(product_prod(A,nat),bool)),Uu),Uua),Uub))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uub)),Uua)),Uu)) ) ).

% ATP.lambda_439
tff(fact_7981_ATP_Olambda__440,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),aTP_Lamp_aey(nat,fun(nat,fun(set(nat),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(set(nat)),bool,aa(set(nat),fun(set(set(nat)),bool),member(set(nat)),Uub),pow2(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu))))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_440
tff(fact_7982_ATP_Olambda__441,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_ky(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)))
        & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uua)) ) ) ).

% ATP.lambda_441
tff(fact_7983_ATP_Olambda__442,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
      ( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_np(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua)))
        & pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ) ).

% ATP.lambda_442
tff(fact_7984_ATP_Olambda__443,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ng(list(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),aa(list(A),set(A),set2(A),Uu)))
            & pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_443
tff(fact_7985_ATP_Olambda__444,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jk(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_444
tff(fact_7986_ATP_Olambda__445,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_abd(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),field2(A,Uu)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uua))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4)),Uu)) ) ) ) ).

% ATP.lambda_445
tff(fact_7987_ATP_Olambda__446,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_abe(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),field2(A,Uu)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uua))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub)),Uu)) ) ) ) ).

% ATP.lambda_446
tff(fact_7988_ATP_Olambda__447,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_abc(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),field2(A,Uu)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uua))
           => ( ( Uub != X4 )
              & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4)),Uu)) ) ) ) ) ).

% ATP.lambda_447
tff(fact_7989_ATP_Olambda__448,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aef(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),field2(A,Uu)))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uua))
           => ( ( Uub != X4 )
              & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub)),Uu)) ) ) ) ) ).

% ATP.lambda_448
tff(fact_7990_ATP_Olambda__449,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_kx(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu))),Uua)) ) ).

% ATP.lambda_449
tff(fact_7991_ATP_Olambda__450,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aaw(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uub)) ) ) ).

% ATP.lambda_450
tff(fact_7992_ATP_Olambda__451,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( pp(aa(A,bool,aa(nat,fun(A,bool),aTP_Lamp_iw(set(A),fun(nat,fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),infini527867602293511546merate(A,Uu,Uua)),Uub)) ) ) ) ).

% ATP.lambda_451
tff(fact_7993_ATP_Olambda__452,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ah(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(nat,nat,suc,Uua))) ) ) ).

% ATP.lambda_452
tff(fact_7994_ATP_Olambda__453,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hc(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_453
tff(fact_7995_ATP_Olambda__454,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_454
tff(fact_7996_ATP_Olambda__455,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_hh(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uua),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_455
tff(fact_7997_ATP_Olambda__456,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ep(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub)) ).

% ATP.lambda_456
tff(fact_7998_ATP_Olambda__457,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ab(vEBT_VEBT,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,vEBT_vebt_member(Uu),Uub))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uub)) ) ) ).

% ATP.lambda_457
tff(fact_7999_ATP_Olambda__458,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aa(vEBT_VEBT,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,vEBT_vebt_member(Uu),Uub))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_458
tff(fact_8000_ATP_Olambda__459,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aay(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uub),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uub),Uu)) ) ) ).

% ATP.lambda_459
tff(fact_8001_ATP_Olambda__460,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_acf(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu))
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu)) ) ) ).

% ATP.lambda_460
tff(fact_8002_ATP_Olambda__461,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_ml(list(A),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uu),Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_461
tff(fact_8003_ATP_Olambda__462,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_nc(fun(A,nat),fun(list(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uua),Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_462
tff(fact_8004_ATP_Olambda__463,axiom,
    ! [B: $tType,Uu: set(B),Uua: fun(B,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_bf(set(B),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
        & pp(aa(B,bool,Uua,Uub)) ) ) ).

% ATP.lambda_463
tff(fact_8005_ATP_Olambda__464,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aft(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_464
tff(fact_8006_ATP_Olambda__465,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_acw(set(A),fun(fun(A,set(B)),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
        & ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).

% ATP.lambda_465
tff(fact_8007_ATP_Olambda__466,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
            & ( aa(B,A,Uua,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_466
tff(fact_8008_ATP_Olambda__467,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_iv(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_467
tff(fact_8009_ATP_Olambda__468,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ap(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
            & ( aa(B,A,Uua,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_468
tff(fact_8010_ATP_Olambda__469,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_ix(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uua))
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_469
tff(fact_8011_ATP_Olambda__470,axiom,
    ! [A: $tType,B: $tType,Uu: list(product_prod(A,B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_oc(list(product_prod(A,B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(A,option(B),map_of(A,B,Uu),Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_470
tff(fact_8012_ATP_Olambda__471,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ya(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uu)),Uua)) ) ).

% ATP.lambda_471
tff(fact_8013_ATP_Olambda__472,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ib(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_472
tff(fact_8014_ATP_Olambda__473,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ig(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_473
tff(fact_8015_ATP_Olambda__474,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ld(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Uub,Uu)),Uua) ) ).

% ATP.lambda_474
tff(fact_8016_ATP_Olambda__475,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_lb(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_475
tff(fact_8017_ATP_Olambda__476,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_lc(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Uub,Uu)),Uua) ) ).

% ATP.lambda_476
tff(fact_8018_ATP_Olambda__477,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_la(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_477
tff(fact_8019_ATP_Olambda__478,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: B,Uub: C] :
      ( pp(aa(C,bool,aa(B,fun(C,bool),aTP_Lamp_ahv(set(product_prod(B,C)),fun(B,fun(C,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(B,C)),bool,aa(product_prod(B,C),fun(set(product_prod(B,C)),bool),member(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)),Uu)) ) ).

% ATP.lambda_478
tff(fact_8020_ATP_Olambda__479,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_agh(set(product_prod(B,A)),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub)),Uu)) ) ).

% ATP.lambda_479
tff(fact_8021_ATP_Olambda__480,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_in(set(product_prod(A,B)),fun(A,fun(B,bool))),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)),Uu)) ) ).

% ATP.lambda_480
tff(fact_8022_ATP_Olambda__481,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aao(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub)),Uu)) ) ).

% ATP.lambda_481
tff(fact_8023_ATP_Olambda__482,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_aet(set(product_prod(B,A)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(B,A)),bool,aa(product_prod(B,A),fun(set(product_prod(B,A)),bool),member(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uub),Uua)),Uu)) ) ).

% ATP.lambda_482
tff(fact_8024_ATP_Olambda__483,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aib(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)),Uu)) ) ).

% ATP.lambda_483
tff(fact_8025_ATP_Olambda__484,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mj(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Uu),Uub)),Uua) ).

% ATP.lambda_484
tff(fact_8026_ATP_Olambda__485,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa(complex,bool,aa(complex,fun(complex,bool),aTP_Lamp_al(nat,fun(complex,fun(complex,bool)),Uu),Uua),Uub))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uu) = Uua ) ) ).

% ATP.lambda_485
tff(fact_8027_ATP_Olambda__486,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa(complex,bool,aa(nat,fun(complex,bool),aTP_Lamp_ao(complex,fun(nat,fun(complex,bool)),Uu),Uua),Uub))
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uua) = Uu ) ) ).

% ATP.lambda_486
tff(fact_8028_ATP_Olambda__487,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_is(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),Uub))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uub),Uua)) ) ) ) ).

% ATP.lambda_487
tff(fact_8029_ATP_Olambda__488,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gv(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_488
tff(fact_8030_ATP_Olambda__489,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gp(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_489
tff(fact_8031_ATP_Olambda__490,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ui(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,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_490
tff(fact_8032_ATP_Olambda__491,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ou(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_491
tff(fact_8033_ATP_Olambda__492,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_gz(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_492
tff(fact_8034_ATP_Olambda__493,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ey(fun(nat,nat),fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),Uub)) ).

% ATP.lambda_493
tff(fact_8035_ATP_Olambda__494,axiom,
    ! [Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa) )
     => ! [Uu: fun(nat,Aa),Uua: Aa,Uub: nat] : aa(nat,Aa,aa(Aa,fun(nat,Aa),aTP_Lamp_tf(fun(nat,Aa),fun(Aa,fun(nat,Aa)),Uu),Uua),Uub) = aa(Aa,Aa,aa(Aa,fun(Aa,Aa),times_times(Aa),aa(nat,Aa,Uu,Uub)),aa(nat,Aa,aa(Aa,fun(nat,Aa),power_power(Aa),Uua),Uub)) ) ).

% ATP.lambda_494
tff(fact_8036_ATP_Olambda__495,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_eg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_495
tff(fact_8037_ATP_Olambda__496,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fk(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_496
tff(fact_8038_ATP_Olambda__497,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_go(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_497
tff(fact_8039_ATP_Olambda__498,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_498
tff(fact_8040_ATP_Olambda__499,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_eo(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_499
tff(fact_8041_ATP_Olambda__500,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_500
tff(fact_8042_ATP_Olambda__501,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_af(fun(nat,bool),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,Uu,Uub))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_501
tff(fact_8043_ATP_Olambda__502,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),aTP_Lamp_nf(fun(B,A),fun(B,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_502
tff(fact_8044_ATP_Olambda__503,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_vl(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_503
tff(fact_8045_ATP_Olambda__504,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_vn(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_504
tff(fact_8046_ATP_Olambda__505,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_ait(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub))) ) ) ).

% ATP.lambda_505
tff(fact_8047_ATP_Olambda__506,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_vi(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uua,Uub)),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_506
tff(fact_8048_ATP_Olambda__507,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_qu(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = divide_divide(A,aa(C,A,Uu,Uub),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_507
tff(fact_8049_ATP_Olambda__508,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_qh(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = divide_divide(A,aa(C,A,Uu,Uub),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_508
tff(fact_8050_ATP_Olambda__509,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_uy(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = divide_divide(A,aa(C,A,Uu,Uub),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_509
tff(fact_8051_ATP_Olambda__510,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rw(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_510
tff(fact_8052_ATP_Olambda__511,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vo(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_511
tff(fact_8053_ATP_Olambda__512,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wp(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_512
tff(fact_8054_ATP_Olambda__513,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_pe(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,Uu,Uub),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_513
tff(fact_8055_ATP_Olambda__514,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_514
tff(fact_8056_ATP_Olambda__515,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(real,A),Uua: fun(real,A),Uub: real] : aa(real,A,aa(fun(real,A),fun(real,A),aTP_Lamp_aic(fun(real,A),fun(fun(real,A),fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,Uu,Uub)),aa(real,A,Uua,Uub)) ) ).

% ATP.lambda_515
tff(fact_8057_ATP_Olambda__516,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aa(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),aTP_Lamp_afe(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat))),Uu),Uua),Uub) = aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(nat,rat,Uu,Uub)),aa(nat,rat,Uua,Uub)) ).

% ATP.lambda_516
tff(fact_8058_ATP_Olambda__517,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_jo(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_517
tff(fact_8059_ATP_Olambda__518,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ws(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_518
tff(fact_8060_ATP_Olambda__519,axiom,
    ! [A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ahf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_519
tff(fact_8061_ATP_Olambda__520,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_jn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_520
tff(fact_8062_ATP_Olambda__521,axiom,
    ! [A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(D,A),Uua: fun(D,A),Uub: D] : aa(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_pz(fun(D,A),fun(fun(D,A),fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uu,Uub)),aa(D,A,Uua,Uub)) ) ).

% ATP.lambda_521
tff(fact_8063_ATP_Olambda__522,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_wm(fun(D,B),fun(fun(D,B),fun(D,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(D,B,Uu,Uub)),aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_522
tff(fact_8064_ATP_Olambda__523,axiom,
    ! [A: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(D,A),Uua: fun(D,A),Uub: D] : aa(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_wl(fun(D,A),fun(fun(D,A),fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uu,Uub)),aa(D,A,Uua,Uub)) ) ).

% ATP.lambda_523
tff(fact_8065_ATP_Olambda__524,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topological_t2_space(D)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_ru(fun(D,B),fun(fun(D,B),fun(D,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(D,B,Uu,Uub)),aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_524
tff(fact_8066_ATP_Olambda__525,axiom,
    ! [A: $tType,D: $tType] :
      ( ( topological_t2_space(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(D,A),Uua: fun(D,A),Uub: D] : aa(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_rv(fun(D,A),fun(fun(D,A),fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uu,Uub)),aa(D,A,Uua,Uub)) ) ).

% ATP.lambda_525
tff(fact_8067_ATP_Olambda__526,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_rr(fun(D,B),fun(fun(D,B),fun(D,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(D,B,Uu,Uub)),aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_526
tff(fact_8068_ATP_Olambda__527,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(D,A),Uua: fun(D,A),Uub: D] : aa(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_sa(fun(D,A),fun(fun(D,A),fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uu,Uub)),aa(D,A,Uua,Uub)) ) ).

% ATP.lambda_527
tff(fact_8069_ATP_Olambda__528,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ux(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_528
tff(fact_8070_ATP_Olambda__529,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ro(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_529
tff(fact_8071_ATP_Olambda__530,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_cy(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_530
tff(fact_8072_ATP_Olambda__531,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vh(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_531
tff(fact_8073_ATP_Olambda__532,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_pk(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_532
tff(fact_8074_ATP_Olambda__533,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_533
tff(fact_8075_ATP_Olambda__534,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wb(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_534
tff(fact_8076_ATP_Olambda__535,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_we(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu,Uub)) ).

% ATP.lambda_535
tff(fact_8077_ATP_Olambda__536,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ge(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_536
tff(fact_8078_ATP_Olambda__537,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_tt(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_537
tff(fact_8079_ATP_Olambda__538,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_sd(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_538
tff(fact_8080_ATP_Olambda__539,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_acu(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_539
tff(fact_8081_ATP_Olambda__540,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_sc(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uua,Uub)),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_540
tff(fact_8082_ATP_Olambda__541,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_df(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_541
tff(fact_8083_ATP_Olambda__542,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_ce(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_542
tff(fact_8084_ATP_Olambda__543,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_acs(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_543
tff(fact_8085_ATP_Olambda__544,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_acq(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_544
tff(fact_8086_ATP_Olambda__545,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_sl(fun(D,B),fun(fun(D,B),fun(D,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(D,B,Uu,Uub)),aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_545
tff(fact_8087_ATP_Olambda__546,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ja(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_546
tff(fact_8088_ATP_Olambda__547,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_or(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_547
tff(fact_8089_ATP_Olambda__548,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ra(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_548
tff(fact_8090_ATP_Olambda__549,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,product_prod(B,B),aa(A,fun(A,product_prod(B,B)),aTP_Lamp_adf(fun(A,B),fun(A,fun(A,product_prod(B,B))),Uu),Uua),Uub) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uu,Uua)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_549
tff(fact_8091_ATP_Olambda__550,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: C] : aa(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_acc(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(C,B,Uua,Uub)) ).

% ATP.lambda_550
tff(fact_8092_ATP_Olambda__551,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_wk(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_551
tff(fact_8093_ATP_Olambda__552,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_si(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_552
tff(fact_8094_ATP_Olambda__553,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_sj(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_553
tff(fact_8095_ATP_Olambda__554,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aby(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ).

% ATP.lambda_554
tff(fact_8096_ATP_Olambda__555,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_agn(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(A,bool,Uu,Uub))
            | pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_555
tff(fact_8097_ATP_Olambda__556,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_ahw(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_556
tff(fact_8098_ATP_Olambda__557,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_aac(B,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uua,Uu) = aa(B,A,Uua,Uub) ) ) ) ).

% ATP.lambda_557
tff(fact_8099_ATP_Olambda__558,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_ea(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uua,Uub))) ) ).

% ATP.lambda_558
tff(fact_8100_ATP_Olambda__559,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_wj(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & ! [Y5: A] :
            ( pp(aa(A,bool,Uua,Y5))
           => pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Y5)) ) ) ) ).

% ATP.lambda_559
tff(fact_8101_ATP_Olambda__560,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_pn(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu,Uub)),aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_560
tff(fact_8102_ATP_Olambda__561,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: fun(A,real),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ajl(fun(A,real),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(real,fun(B,B),real_V8093663219630862766scaleR(B),aa(A,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_561
tff(fact_8103_ATP_Olambda__562,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_sg(set(B),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_562
tff(fact_8104_ATP_Olambda__563,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ag(fun(nat,nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua)) ) ).

% ATP.lambda_563
tff(fact_8105_ATP_Olambda__564,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_vk(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_564
tff(fact_8106_ATP_Olambda__565,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_vr(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_565
tff(fact_8107_ATP_Olambda__566,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_wi(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_566
tff(fact_8108_ATP_Olambda__567,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_wh(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_567
tff(fact_8109_ATP_Olambda__568,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gk(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uu,Uub),Uua) ) ).

% ATP.lambda_568
tff(fact_8110_ATP_Olambda__569,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_rx(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_569
tff(fact_8111_ATP_Olambda__570,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_tl(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uua,Uub),Uu) ) ).

% ATP.lambda_570
tff(fact_8112_ATP_Olambda__571,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_vu(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_571
tff(fact_8113_ATP_Olambda__572,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(real,A),Uua: A,Uub: real] : aa(real,A,aa(A,fun(real,A),aTP_Lamp_aie(fun(real,A),fun(A,fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_572
tff(fact_8114_ATP_Olambda__573,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ck(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_573
tff(fact_8115_ATP_Olambda__574,axiom,
    ! [D: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(D,A),Uua: A,Uub: D] : aa(D,A,aa(A,fun(D,A),aTP_Lamp_rz(fun(D,A),fun(A,fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_574
tff(fact_8116_ATP_Olambda__575,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(C,A),Uua: A,Uub: C] : aa(C,A,aa(A,fun(C,A),aTP_Lamp_pw(fun(C,A),fun(A,fun(C,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_575
tff(fact_8117_ATP_Olambda__576,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_vg(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_576
tff(fact_8118_ATP_Olambda__577,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_wo(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_577
tff(fact_8119_ATP_Olambda__578,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(A)
        & topological_t2_space(B) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_rs(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_578
tff(fact_8120_ATP_Olambda__579,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_rq(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_579
tff(fact_8121_ATP_Olambda__580,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_bc(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_580
tff(fact_8122_ATP_Olambda__581,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_pi(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_581
tff(fact_8123_ATP_Olambda__582,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qn(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),Uu) ) ).

% ATP.lambda_582
tff(fact_8124_ATP_Olambda__583,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_co(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_583
tff(fact_8125_ATP_Olambda__584,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_tj(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_584
tff(fact_8126_ATP_Olambda__585,axiom,
    ! [B: $tType,A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rk(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_585
tff(fact_8127_ATP_Olambda__586,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_586
tff(fact_8128_ATP_Olambda__587,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_sb(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_587
tff(fact_8129_ATP_Olambda__588,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_ol(fun(real,real),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uu,Uub)),Uua) ).

% ATP.lambda_588
tff(fact_8130_ATP_Olambda__589,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_qq(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_589
tff(fact_8131_ATP_Olambda__590,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_po(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_590
tff(fact_8132_ATP_Olambda__591,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_uz(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_591
tff(fact_8133_ATP_Olambda__592,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_sr(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_592
tff(fact_8134_ATP_Olambda__593,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wg(nat,fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uua,Uub)),Uu) ).

% ATP.lambda_593
tff(fact_8135_ATP_Olambda__594,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rm(nat,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uua,Uub)),Uu) ).

% ATP.lambda_594
tff(fact_8136_ATP_Olambda__595,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_xq(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_595
tff(fact_8137_ATP_Olambda__596,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_wv(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_596
tff(fact_8138_ATP_Olambda__597,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_oq(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_597
tff(fact_8139_ATP_Olambda__598,axiom,
    ! [E: $tType,F: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(E,fun(F,product_prod(C,D))),Uua: fun(C,fun(D,B)),Uub: E] : aa(E,fun(F,B),aa(fun(C,fun(D,B)),fun(E,fun(F,B)),aTP_Lamp_afk(fun(E,fun(F,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F,B))),Uu),Uua),Uub) = product_scomp(F,C,D,B,aa(E,fun(F,product_prod(C,D)),Uu,Uub),Uua) ).

% ATP.lambda_598
tff(fact_8140_ATP_Olambda__599,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(C,A),Uua: int,Uub: C] : aa(C,A,aa(int,fun(C,A),aTP_Lamp_zq(fun(C,A),fun(int,fun(C,A)),Uu),Uua),Uub) = power_int(A,aa(C,A,Uu,Uub),Uua) ) ).

% ATP.lambda_599
tff(fact_8141_ATP_Olambda__600,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(B,A),Uua: int,Uub: B] : aa(B,A,aa(int,fun(B,A),aTP_Lamp_zk(fun(B,A),fun(int,fun(B,A)),Uu),Uua),Uub) = power_int(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_600
tff(fact_8142_ATP_Olambda__601,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : aa(A,B,aa(int,fun(A,B),aTP_Lamp_zn(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_601
tff(fact_8143_ATP_Olambda__602,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : aa(A,B,aa(int,fun(A,B),aTP_Lamp_zj(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_602
tff(fact_8144_ATP_Olambda__603,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_zp(fun(A,A),fun(int,fun(A,A)),Uu),Uua),Uub) = power_int(A,aa(A,A,Uu,Uub),Uua) ) ).

% ATP.lambda_603
tff(fact_8145_ATP_Olambda__604,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : aa(A,B,aa(int,fun(A,B),aTP_Lamp_zl(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_604
tff(fact_8146_ATP_Olambda__605,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,option(B)),Uua: fun(B,option(A)),Uub: C] : aa(C,option(A),aa(fun(B,option(A)),fun(C,option(A)),aTP_Lamp_ahg(fun(C,option(B)),fun(fun(B,option(A)),fun(C,option(A))),Uu),Uua),Uub) = aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),aa(C,option(B),Uu,Uub)),Uua) ).

% ATP.lambda_605
tff(fact_8147_ATP_Olambda__606,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_agl(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(B,A,Uu,Uub) = Uua ) ) ).

% ATP.lambda_606
tff(fact_8148_ATP_Olambda__607,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_afi(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub != Uua )
        & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub)),Uu)) ) ) ).

% ATP.lambda_607
tff(fact_8149_ATP_Olambda__608,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_adv(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub != Uua )
        & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)),Uu)) ) ) ).

% ATP.lambda_608
tff(fact_8150_ATP_Olambda__609,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jj(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub))) ) ).

% ATP.lambda_609
tff(fact_8151_ATP_Olambda__610,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(nat,A),Uub: nat] : aa(nat,real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_ha(real,fun(fun(nat,A),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uua,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_610
tff(fact_8152_ATP_Olambda__611,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_eb(fun(B,bool),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uub))),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_611
tff(fact_8153_ATP_Olambda__612,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_agj(fun(A,A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( ? [X4: A] :
                ( ( Uub = aa(A,A,Uu,X4) )
                & pp(aa(A,bool,Uua,X4)) )
            | ? [M9: set(A)] :
                ( ( Uub = aa(set(A),A,complete_Sup_Sup(A),M9) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M9)
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),M9))
                   => pp(aa(A,bool,Uua,X4)) ) ) ) ) ) ).

% ATP.lambda_612
tff(fact_8154_ATP_Olambda__613,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_afz(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),Uua))
        & pp(aa(set(A),bool,finite_finite2(A),Uub))
        & ( Uub != bot_bot(set(A)) )
        & ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uua))
           => ? [Xa3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),Uub))
                & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),Uu)) ) ) ) ) ).

% ATP.lambda_613
tff(fact_8155_ATP_Olambda__614,axiom,
    ! [A: $tType,Uu: fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),Uua: fun(bool,fun(bool,A)),Uub: vEBT_VEBT] : aa(vEBT_VEBT,product_prod(vEBT_VEBT,A),aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A)),aTP_Lamp_adx(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,product_prod(vEBT_VEBT,A))),Uu),Uua),Uub) = aa(A,product_prod(vEBT_VEBT,A),aa(vEBT_VEBT,fun(A,product_prod(vEBT_VEBT,A)),product_Pair(vEBT_VEBT,A),Uub),aa(vEBT_VEBT,A,aa(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(product_prod(vEBT_VEBT,A)),fun(vEBT_VEBT,fun(A,A))))),fun(fun(bool,fun(bool,A)),fun(vEBT_VEBT,A)),vEBT_rec_VEBT(A),Uu),Uua),Uub)) ).

% ATP.lambda_614
tff(fact_8156_ATP_Olambda__615,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jg(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_615
tff(fact_8157_ATP_Olambda__616,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ji(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_616
tff(fact_8158_ATP_Olambda__617,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jf(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_617
tff(fact_8159_ATP_Olambda__618,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_zi(A,fun(int,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Uua)),power_int(A,Uu,aa(int,int,aa(int,fun(int,int),minus_minus(int),Uua),one_one(int))))) ) ).

% ATP.lambda_618
tff(fact_8160_ATP_Olambda__619,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bx(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_619
tff(fact_8161_ATP_Olambda__620,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bv(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_620
tff(fact_8162_ATP_Olambda__621,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ds(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_621
tff(fact_8163_ATP_Olambda__622,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bm(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_622
tff(fact_8164_ATP_Olambda__623,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_bw(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ).

% ATP.lambda_623
tff(fact_8165_ATP_Olambda__624,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: A,Uua: B,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(B,fun(C,product_prod(A,product_prod(B,C))),aa(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_ms(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uu),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)) ).

% ATP.lambda_624
tff(fact_8166_ATP_Olambda__625,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: B,Uua: A,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(A,fun(C,product_prod(A,product_prod(B,C))),aTP_Lamp_mt(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uua),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uu),Uub)) ).

% ATP.lambda_625
tff(fact_8167_ATP_Olambda__626,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
          ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_me(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),image2(A,B,Uu,Uua))) ) ) ).

% ATP.lambda_626
tff(fact_8168_ATP_Olambda__627,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_vt(A,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_627
tff(fact_8169_ATP_Olambda__628,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_vj(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_628
tff(fact_8170_ATP_Olambda__629,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_vv(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_629
tff(fact_8171_ATP_Olambda__630,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_vm(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_630
tff(fact_8172_ATP_Olambda__631,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_vs(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_631
tff(fact_8173_ATP_Olambda__632,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ob(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_632
tff(fact_8174_ATP_Olambda__633,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cp(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_633
tff(fact_8175_ATP_Olambda__634,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vf(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_634
tff(fact_8176_ATP_Olambda__635,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bb(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_635
tff(fact_8177_ATP_Olambda__636,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cn(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_636
tff(fact_8178_ATP_Olambda__637,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_tk(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_637
tff(fact_8179_ATP_Olambda__638,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rl(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_638
tff(fact_8180_ATP_Olambda__639,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(real,A),Uua: A,Uub: real] : aa(real,A,aa(A,fun(real,A),aTP_Lamp_aid(fun(real,A),fun(A,fun(real,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(real,A,Uu,Uub)) ) ).

% ATP.lambda_639
tff(fact_8181_ATP_Olambda__640,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cl(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_640
tff(fact_8182_ATP_Olambda__641,axiom,
    ! [A: $tType,D: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(D,A),Uua: A,Uub: D] : aa(D,A,aa(A,fun(D,A),aTP_Lamp_ry(fun(D,A),fun(A,fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(D,A,Uu,Uub)) ) ).

% ATP.lambda_641
tff(fact_8183_ATP_Olambda__642,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(C,A),Uua: A,Uub: C] : aa(C,A,aa(A,fun(C,A),aTP_Lamp_pv(fun(C,A),fun(A,fun(C,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(C,A,Uu,Uub)) ) ).

% ATP.lambda_642
tff(fact_8184_ATP_Olambda__643,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_wn(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_643
tff(fact_8185_ATP_Olambda__644,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_rt(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_644
tff(fact_8186_ATP_Olambda__645,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_rp(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_645
tff(fact_8187_ATP_Olambda__646,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_vw(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_646
tff(fact_8188_ATP_Olambda__647,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_pj(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_647
tff(fact_8189_ATP_Olambda__648,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(B,nat),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_uc(fun(B,nat),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(B,nat,Uu,Uub)) ) ).

% ATP.lambda_648
tff(fact_8190_ATP_Olambda__649,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xo(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_649
tff(fact_8191_ATP_Olambda__650,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_wx(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_650
tff(fact_8192_ATP_Olambda__651,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,B),Uua: A,Uub: C] : aa(C,product_prod(A,B),aa(A,fun(C,product_prod(A,B)),aTP_Lamp_lz(fun(C,B),fun(A,fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(C,B,Uu,Uub)) ).

% ATP.lambda_651
tff(fact_8193_ATP_Olambda__652,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_nm(fun(list(A),A),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_652
tff(fact_8194_ATP_Olambda__653,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_dr(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_653
tff(fact_8195_ATP_Olambda__654,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: set(A),Uua: set(C),Uub: fun(A,B)] : aa(fun(A,B),set(fun(A,C)),aa(set(C),fun(fun(A,B),set(fun(A,C))),aTP_Lamp_ajx(set(A),fun(set(C),fun(fun(A,B),set(fun(A,C)))),Uu),Uua),Uub) = bNF_Wellorder_Func(A,C,Uu,Uua) ).

% ATP.lambda_654
tff(fact_8196_ATP_Olambda__655,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_acr(set(B),fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Uu),Uua) ).

% ATP.lambda_655
tff(fact_8197_ATP_Olambda__656,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,set(A),aa(A,fun(A,set(A)),aTP_Lamp_ajs(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_underS(A,Uu,Uua) ).

% ATP.lambda_656
tff(fact_8198_ATP_Olambda__657,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(B),Uua: set(C),Uub: A] : aa(A,set(sum_sum(B,C)),aa(set(C),fun(A,set(sum_sum(B,C))),aTP_Lamp_aju(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),Uu),Uua),Uub) = sum_Plus(B,C,Uu,Uua) ).

% ATP.lambda_657
tff(fact_8199_ATP_Olambda__658,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_acn(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),insert(B,Uu),Uua) ).

% ATP.lambda_658
tff(fact_8200_ATP_Olambda__659,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,B),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_ade(fun(D,B),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = image2(D,B,Uu,Uua) ).

% ATP.lambda_659
tff(fact_8201_ATP_Olambda__660,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: fun(A,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(A,bool),fun(B,bool),aTP_Lamp_agm(fun(A,B),fun(fun(A,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,Uua,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),Uu),Uub))) ) ).

% ATP.lambda_660
tff(fact_8202_ATP_Olambda__661,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_cx(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_661
tff(fact_8203_ATP_Olambda__662,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_de(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_662
tff(fact_8204_ATP_Olambda__663,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aiz(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_663
tff(fact_8205_ATP_Olambda__664,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_tq(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ) ).

% ATP.lambda_664
tff(fact_8206_ATP_Olambda__665,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_sq(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_665
tff(fact_8207_ATP_Olambda__666,axiom,
    ! [Uu: fun(nat,real),Uua: nat,Uub: nat] : aa(nat,real,aa(nat,fun(nat,real),aTP_Lamp_ahe(fun(nat,real),fun(nat,fun(nat,real)),Uu),Uua),Uub) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_666
tff(fact_8208_ATP_Olambda__667,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_xb(fun(nat,set(A)),fun(nat,fun(nat,set(A))),Uu),Uua),Uub) = aa(nat,set(A),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_667
tff(fact_8209_ATP_Olambda__668,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gq(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_668
tff(fact_8210_ATP_Olambda__669,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_dc(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_669
tff(fact_8211_ATP_Olambda__670,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bl(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_670
tff(fact_8212_ATP_Olambda__671,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aje(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_671
tff(fact_8213_ATP_Olambda__672,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,bool),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_vq(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua))) ) ) ).

% ATP.lambda_672
tff(fact_8214_ATP_Olambda__673,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_tb(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_673
tff(fact_8215_ATP_Olambda__674,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),bool),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_afx(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(A,B),bool,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub))) ) ).

% ATP.lambda_674
tff(fact_8216_ATP_Olambda__675,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_hw(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_675
tff(fact_8217_ATP_Olambda__676,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),bool),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_abu(fun(product_prod(A,A),bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(product_prod(A,A),bool,Uu,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua))) ) ) ).

% ATP.lambda_676
tff(fact_8218_ATP_Olambda__677,axiom,
    ! [D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D) )
     => ! [Uu: A,Uua: fun(A,D),Uub: A] : aa(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_ss(A,fun(fun(A,D),fun(A,D)),Uu),Uua),Uub) = aa(A,D,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uub)) ) ).

% ATP.lambda_677
tff(fact_8219_ATP_Olambda__678,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_so(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uub)) ) ).

% ATP.lambda_678
tff(fact_8220_ATP_Olambda__679,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cd(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uu)) ) ).

% ATP.lambda_679
tff(fact_8221_ATP_Olambda__680,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qm(fun(real,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,Uu,aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_680
tff(fact_8222_ATP_Olambda__681,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,nat),Uub: nat] : aa(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_abi(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_681
tff(fact_8223_ATP_Olambda__682,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(B,bool)),Uua: fun(A,C),Uub: A] : aa(A,fun(B,bool),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_ahl(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Uu),Uua),Uub) = aa(C,fun(B,bool),Uu,aa(A,C,Uua,Uub)) ).

% ATP.lambda_682
tff(fact_8224_ATP_Olambda__683,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,B),Uub: A] : aa(A,C,aa(fun(A,B),fun(A,C),aTP_Lamp_yd(fun(B,C),fun(fun(A,B),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(A,B,Uua,Uub)) ).

% ATP.lambda_683
tff(fact_8225_ATP_Olambda__684,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aga(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_684
tff(fact_8226_ATP_Olambda__685,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ael(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_685
tff(fact_8227_ATP_Olambda__686,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_up(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_686
tff(fact_8228_ATP_Olambda__687,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_pm(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_687
tff(fact_8229_ATP_Olambda__688,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_afq(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_688
tff(fact_8230_ATP_Olambda__689,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xf(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_689
tff(fact_8231_ATP_Olambda__690,axiom,
    ! [B: $tType,C: $tType,A: $tType,D: $tType,Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: D] : aa(D,fun(B,C),aa(fun(D,A),fun(D,fun(B,C)),aTP_Lamp_jz(fun(A,fun(B,C)),fun(fun(D,A),fun(D,fun(B,C))),Uu),Uua),Uub) = aa(A,fun(B,C),Uu,aa(D,A,Uua,Uub)) ).

% ATP.lambda_690
tff(fact_8232_ATP_Olambda__691,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(num,A),Uub: num] : aa(num,B,aa(fun(num,A),fun(num,B),aTP_Lamp_aff(fun(A,B),fun(fun(num,A),fun(num,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(num,A,Uua,Uub)) ).

% ATP.lambda_691
tff(fact_8233_ATP_Olambda__692,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_jr(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ).

% ATP.lambda_692
tff(fact_8234_ATP_Olambda__693,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,nat),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_abj(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_693
tff(fact_8235_ATP_Olambda__694,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,nat),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_abk(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_694
tff(fact_8236_ATP_Olambda__695,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_iu(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_695
tff(fact_8237_ATP_Olambda__696,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_pl(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_696
tff(fact_8238_ATP_Olambda__697,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_1(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_le(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_697
tff(fact_8239_ATP_Olambda__698,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: D,Uub: A] : aa(A,set(B),aa(D,fun(A,set(B)),aTP_Lamp_acx(fun(D,set(B)),fun(D,fun(A,set(B))),Uu),Uua),Uub) = aa(D,set(B),Uu,Uua) ).

% ATP.lambda_698
tff(fact_8240_ATP_Olambda__699,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),bool),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),bool),aa(A,fun(A,fun(product_prod(A,A),bool)),aTP_Lamp_abw(fun(product_prod(A,A),bool),fun(A,fun(A,fun(product_prod(A,A),bool))),Uu),Uua),Uub) = aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_abv(fun(product_prod(A,A),bool),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)) ) ).

% ATP.lambda_699
tff(fact_8241_ATP_Olambda__700,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),bool),aa(A,fun(A,fun(product_prod(A,A),bool)),aTP_Lamp_abm(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),Uu),Uua),Uub) = aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_abl(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)) ).

% ATP.lambda_700
tff(fact_8242_ATP_Olambda__701,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa) )
     => ! [Uu: fun(A,Aa),Uua: fun(nat,Aa),Uub: A] : aa(A,Aa,aa(fun(nat,Aa),fun(A,Aa),aTP_Lamp_th(fun(A,Aa),fun(fun(nat,Aa),fun(A,Aa)),Uu),Uua),Uub) = suminf(Aa,aa(A,fun(nat,Aa),aa(fun(nat,Aa),fun(A,fun(nat,Aa)),aTP_Lamp_tg(fun(A,Aa),fun(fun(nat,Aa),fun(A,fun(nat,Aa))),Uu),Uua),Uub)) ) ).

% ATP.lambda_701
tff(fact_8243_ATP_Olambda__702,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_rj(fun(nat,A),fun(A,fun(A,A)),Uu),Uua),Uub) = suminf(A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_ri(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_702
tff(fact_8244_ATP_Olambda__703,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_ail(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),Uu),Uua),Uub) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uua),aa(A,fun(A,A),Uu,Uua)),Uub)) ).

% ATP.lambda_703
tff(fact_8245_ATP_Olambda__704,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: A] : aa(A,option(A),aa(A,fun(A,option(A)),aTP_Lamp_ahj(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uua),Uub) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Uu,Uua),Uub)) ).

% ATP.lambda_704
tff(fact_8246_ATP_Olambda__705,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_agf(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( real_V7696804695334737415tation(A,real_V4986007116245087402_basis(A,Uu),Uua,Uub) != zero_zero(real) ) ) ) ).

% ATP.lambda_705
tff(fact_8247_ATP_Olambda__706,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_gn(fun(nat,A),fun(A,fun(nat,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub))) ) ).

% ATP.lambda_706
tff(fact_8248_ATP_Olambda__707,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_uh(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = cos(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),Uua)) ).

% ATP.lambda_707
tff(fact_8249_ATP_Olambda__708,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_ada(fun(D,set(B)),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),image2(D,set(B),Uu,Uua)) ).

% ATP.lambda_708
tff(fact_8250_ATP_Olambda__709,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_ci(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_ch(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub))) ).

% ATP.lambda_709
tff(fact_8251_ATP_Olambda__710,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & archim2362893244070406136eiling(Aa)
        & topolo2564578578187576103pology(Aa) )
     => ! [Uu: fun(A,real),Uua: fun(real,Aa),Uub: A] : aa(A,real,aa(fun(real,Aa),fun(A,real),aTP_Lamp_rg(fun(A,real),fun(fun(real,Aa),fun(A,real)),Uu),Uua),Uub) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(Aa,aa(real,Aa,Uua,aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_710
tff(fact_8252_ATP_Olambda__711,axiom,
    ! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_zb(list(A),fun(list(B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [I: nat] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Uu),I)),aa(nat,B,nth(B,Uua),I)) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(list(B),nat,size_size(list(B)),Uua)))) ) ) ).

% ATP.lambda_711
tff(fact_8253_ATP_Olambda__712,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_aia(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [A7: A] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),aa(A,B,Uua,A7)) )
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A7),Uu)) ) ) ).

% ATP.lambda_712
tff(fact_8254_ATP_Olambda__713,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_yl(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [I: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu),I) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Uu)))
          & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),I),Uua)) ) ) ).

% ATP.lambda_713
tff(fact_8255_ATP_Olambda__714,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ahz(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ! [X4: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),X4),Uu))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_ahy(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)),X4)) ) ) ).

% ATP.lambda_714
tff(fact_8256_ATP_Olambda__715,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_agg(set(product_prod(A,B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uua))
          & pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Uub)),Uu)) ) ) ).

% ATP.lambda_715
tff(fact_8257_ATP_Olambda__716,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_agb(fun(A,option(B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uua))
          & ( aa(A,option(B),Uu,X4) = aa(B,option(B),some(B),Uub) ) ) ) ).

% ATP.lambda_716
tff(fact_8258_ATP_Olambda__717,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_wa(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uub),N5))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or7035219750837199246ssThan(nat,Uub,N5)))),aa(nat,real,Uua,Uub))) ) ) ) ).

% ATP.lambda_717
tff(fact_8259_ATP_Olambda__718,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_vz(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> ! [A7: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uub),A7))
             => ! [B5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A7),B5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or3652927894154168847AtMost(nat,A7,B5)))),aa(nat,real,Uua,A7))) ) ) ) ) ).

% ATP.lambda_718
tff(fact_8260_ATP_Olambda__719,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ym(fun(A,A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ? [N5: nat] : Uub = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N5),Uu),Uua) ) ).

% ATP.lambda_719
tff(fact_8261_ATP_Olambda__720,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aa(fun(B,A),fun(product_prod(A,A),bool),aTP_Lamp_aer(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),Uu),Uua),Uub))
    <=> ? [A17: B,A26: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uua,A17)),aa(B,A,Uua,A26)) )
          & pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A17),A26)),Uu)) ) ) ).

% ATP.lambda_720
tff(fact_8262_ATP_Olambda__721,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_yu(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [A7: A,V4: list(A)] :
          ( ( Uub = append(A,Uua,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),V4)) )
          | ? [U5: list(A),Aa4: A,B5: A,Va4: list(A),W4: list(A)] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa4),B5)),Uu))
              & ( Uua = append(A,U5,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa4),Va4)) )
              & ( Uub = append(A,U5,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B5),W4)) ) ) ) ) ).

% ATP.lambda_721
tff(fact_8263_ATP_Olambda__722,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_yi(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [Us3: list(A),Z3: A,Z8: A,Vs3: list(A)] :
          ( ( Uua = append(A,Us3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z3),Vs3)) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Z8)),Uu))
          & ( Uub = append(A,Us3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z8),Vs3)) ) ) ) ).

% ATP.lambda_722
tff(fact_8264_ATP_Olambda__723,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_ajk(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),aTP_Lamp_ajj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uub),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc)) ).

% ATP.lambda_723
tff(fact_8265_ATP_Olambda__724,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_ajd(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),Uub,Uuc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uuc)) ).

% ATP.lambda_724
tff(fact_8266_ATP_Olambda__725,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_aja(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),Uub,Uuc)),aa(nat,set(nat),set_ord_lessThan(nat),Uuc)) ).

% ATP.lambda_725
tff(fact_8267_ATP_Olambda__726,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hp(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,fconj(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,bool,aa(nat,fun(nat,bool),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,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),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,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_726
tff(fact_8268_ATP_Olambda__727,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hi(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,fconj(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),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,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,uminus_uminus(real),divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),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,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_727
tff(fact_8269_ATP_Olambda__728,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hk(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),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,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_728
tff(fact_8270_ATP_Olambda__729,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: product_prod(C,A),Uua: A,Uub: B,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_aaa(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = if(set(product_prod(C,B)),aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(product_prod(C,A),A,product_snd(C,A),Uu)),Uua),aa(set(product_prod(C,B)),set(product_prod(C,B)),insert(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).

% ATP.lambda_729
tff(fact_8271_ATP_Olambda__730,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),zero_zero(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% ATP.lambda_730
tff(fact_8272_ATP_Olambda__731,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fe(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),one_one(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% ATP.lambda_731
tff(fact_8273_ATP_Olambda__732,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_vy(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uuc),Uu),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_732
tff(fact_8274_ATP_Olambda__733,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ff(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_733
tff(fact_8275_ATP_Olambda__734,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_734
tff(fact_8276_ATP_Olambda__735,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_aiy(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ).

% ATP.lambda_735
tff(fact_8277_ATP_Olambda__736,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_di(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_736
tff(fact_8278_ATP_Olambda__737,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_dj(B,fun(fun(B,A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),Uub) ) ).

% ATP.lambda_737
tff(fact_8279_ATP_Olambda__738,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_fq(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_738
tff(fact_8280_ATP_Olambda__739,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: C,Uub: A,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(A,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_aag(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_aaf(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_739
tff(fact_8281_ATP_Olambda__740,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: A,Uub: B,Uuc: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(B,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_zz(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(B,C),set(product_prod(A,C)),aa(fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(B,C),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(B,C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_zy(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_740
tff(fact_8282_ATP_Olambda__741,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_kg(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uuc),aa(A,fun(A,A),times_times(A),Uua)),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))) ) ).

% ATP.lambda_741
tff(fact_8283_ATP_Olambda__742,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: fun(D,B),Uuc: D] : aa(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_aek(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,Uuc)),aa(D,B,Uub,Uuc)) ) ).

% ATP.lambda_742
tff(fact_8284_ATP_Olambda__743,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: B,Uuc: D] : aa(D,C,aa(B,fun(D,C),aa(fun(D,A),fun(B,fun(D,C)),aTP_Lamp_aej(fun(A,fun(B,C)),fun(fun(D,A),fun(B,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,Uuc)),Uub) ) ).

% ATP.lambda_743
tff(fact_8285_ATP_Olambda__744,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,B),Uub: A,Uuc: D] : aa(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_aei(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),aa(D,B,Uua,Uuc)) ) ).

% ATP.lambda_744
tff(fact_8286_ATP_Olambda__745,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,B),Uub: A,Uuc: D] : aa(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_agk(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),aa(D,B,Uua,Uuc)) ) ).

% ATP.lambda_745
tff(fact_8287_ATP_Olambda__746,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,C),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_ahm(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(C,bool,aa(A,fun(C,bool),Uu,Uub),aa(B,C,Uua,Uuc))) ) ).

% ATP.lambda_746
tff(fact_8288_ATP_Olambda__747,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_cu(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(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_ct(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(nat,nat,suc,zero_zero(nat)))),Uuc))) ) ).

% ATP.lambda_747
tff(fact_8289_ATP_Olambda__748,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,B,aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_be(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_bd(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_748
tff(fact_8290_ATP_Olambda__749,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_fu(nat,fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),zero_zero(nat)),aa(A,A,uminus_uminus(A),Uub),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),one_one(A),zero_zero(A)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_749
tff(fact_8291_ATP_Olambda__750,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_age(set(A),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(real,fun(B,B),real_V8093663219630862766scaleR(B),real_V7696804695334737415tation(A,real_V4986007116245087402_basis(A,Uu),Uub,Uuc)),if(B,aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu),aa(A,B,Uua,Uuc),zero_zero(B))) ) ).

% ATP.lambda_750
tff(fact_8292_ATP_Olambda__751,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_fa(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(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_ez(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uub),Uuc)) ).

% ATP.lambda_751
tff(fact_8293_ATP_Olambda__752,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_ev(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_eu(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_752
tff(fact_8294_ATP_Olambda__753,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_pc(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Uua)),Uuc)),zero_zero(real)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_753
tff(fact_8295_ATP_Olambda__754,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_pa(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)),zero_zero(real)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_754
tff(fact_8296_ATP_Olambda__755,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_oz(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uua),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uub),Uua)),Uuc)) ).

% ATP.lambda_755
tff(fact_8297_ATP_Olambda__756,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_oy(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uub),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uua),Uub)),Uuc)) ).

% ATP.lambda_756
tff(fact_8298_ATP_Olambda__757,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_dl(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_757
tff(fact_8299_ATP_Olambda__758,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: set(A),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_adu(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),product_Sigma(A,A,Uua,aTP_Lamp_acv(set(A),fun(A,set(A)),Uua))))
        & pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uuc)) ) ) ).

% ATP.lambda_758
tff(fact_8300_ATP_Olambda__759,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_kj(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uub))) ).

% ATP.lambda_759
tff(fact_8301_ATP_Olambda__760,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ct(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Uuc))) ) ).

% ATP.lambda_760
tff(fact_8302_ATP_Olambda__761,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ue(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),aa(nat,A,Uub,Uuc)))),aa(A,A,Uu,Uua)),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_761
tff(fact_8303_ATP_Olambda__762,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_yv(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),Uua)) ) ) ) ).

% ATP.lambda_762
tff(fact_8304_ATP_Olambda__763,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_afc(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uub)),aa(A,B,Uua,Uuc))),Uu)) ) ).

% ATP.lambda_763
tff(fact_8305_ATP_Olambda__764,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ex(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_764
tff(fact_8306_ATP_Olambda__765,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_et(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,semiring_1_of_nat(nat),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_765
tff(fact_8307_ATP_Olambda__766,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ew(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_766
tff(fact_8308_ATP_Olambda__767,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hm(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(A,A,aa(real,fun(A,A),real_V8093663219630862766scaleR(A),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_767
tff(fact_8309_ATP_Olambda__768,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_yt(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys2: list(A),X4: A,Y5: A,Xs5: list(A),Ys6: list(A)] :
            ( ( Uub = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs5)) )
            & ( Uuc = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys6)) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5)),Uu)) ) ) ) ).

% ATP.lambda_768
tff(fact_8310_ATP_Olambda__769,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_do(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)) ) ).

% ATP.lambda_769
tff(fact_8311_ATP_Olambda__770,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_ch(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_770
tff(fact_8312_ATP_Olambda__771,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,bool)),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_cg(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uuc),Uub)) ) ) ).

% ATP.lambda_771
tff(fact_8313_ATP_Olambda__772,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_dm(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_772
tff(fact_8314_ATP_Olambda__773,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_lf(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_773
tff(fact_8315_ATP_Olambda__774,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dn(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uuc))) ) ).

% ATP.lambda_774
tff(fact_8316_ATP_Olambda__775,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_eq(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_775
tff(fact_8317_ATP_Olambda__776,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_li(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ) ).

% ATP.lambda_776
tff(fact_8318_ATP_Olambda__777,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_km(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_777
tff(fact_8319_ATP_Olambda__778,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_lk(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ) ).

% ATP.lambda_778
tff(fact_8320_ATP_Olambda__779,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_ko(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)) ).

% ATP.lambda_779
tff(fact_8321_ATP_Olambda__780,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_kq(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ).

% ATP.lambda_780
tff(fact_8322_ATP_Olambda__781,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: list(A),Uub: B,Uuc: list(B)] : aa(list(B),list(product_prod(A,B)),aa(B,fun(list(B),list(product_prod(A,B))),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_zs(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uub)),zip(A,B,Uua,Uuc)) ).

% ATP.lambda_781
tff(fact_8323_ATP_Olambda__782,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: list(B),Uub: A,Uuc: list(A)] : aa(list(A),list(product_prod(A,B)),aa(A,fun(list(A),list(product_prod(A,B))),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_zt(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uu)),zip(A,B,Uuc,Uua)) ).

% ATP.lambda_782
tff(fact_8324_ATP_Olambda__783,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_ir(A,fun(B,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_783
tff(fact_8325_ATP_Olambda__784,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),aTP_Lamp_xg(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)),aa(C,set(product_prod(A,B)),Uu,Uua))) ) ).

% ATP.lambda_784
tff(fact_8326_ATP_Olambda__785,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_aq(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != one_one(A) ) ) ) ) ).

% ATP.lambda_785
tff(fact_8327_ATP_Olambda__786,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_as(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_786
tff(fact_8328_ATP_Olambda__787,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_um(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(real,fun(B,B),real_V8093663219630862766scaleR(B),divide_divide(real,one_one(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))))) ) ).

% ATP.lambda_787
tff(fact_8329_ATP_Olambda__788,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_gw(fun(nat,A),fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_788
tff(fact_8330_ATP_Olambda__789,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_jm(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_789
tff(fact_8331_ATP_Olambda__790,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_aco(fun(A,bool),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(A,bool,Uu,Uub))
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_790
tff(fact_8332_ATP_Olambda__791,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,Uu: fun(C,set(A)),Uua: fun(D,set(B)),Uub: C,Uuc: D] : aa(D,set(product_prod(A,B)),aa(C,fun(D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_acy(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = product_Sigma(A,B,aa(C,set(A),Uu,Uub),aa(D,fun(A,set(B)),aTP_Lamp_acx(fun(D,set(B)),fun(D,fun(A,set(B))),Uua),Uuc)) ).

% ATP.lambda_791
tff(fact_8333_ATP_Olambda__792,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_ri(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)),Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_792
tff(fact_8334_ATP_Olambda__793,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,Aa),Uua: fun(nat,Aa),Uub: A,Uuc: nat] : aa(nat,Aa,aa(A,fun(nat,Aa),aa(fun(nat,Aa),fun(A,fun(nat,Aa)),aTP_Lamp_tg(fun(A,Aa),fun(fun(nat,Aa),fun(A,fun(nat,Aa))),Uu),Uua),Uub),Uuc) = aa(Aa,Aa,aa(Aa,fun(Aa,Aa),times_times(Aa),aa(nat,Aa,Uua,Uuc)),aa(nat,Aa,aa(Aa,fun(nat,Aa),power_power(Aa),aa(A,Aa,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_793
tff(fact_8335_ATP_Olambda__794,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_rd(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),divide_divide(real,aa(real,real,inverse_inverse(real),sqrt(aa(A,real,Uu,Uua))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% ATP.lambda_794
tff(fact_8336_ATP_Olambda__795,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fl(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_795
tff(fact_8337_ATP_Olambda__796,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: A] : aa(A,A,aa(B,fun(A,A),aa(A,fun(B,fun(A,A)),aTP_Lamp_ny(fun(B,A),fun(A,fun(B,fun(A,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uuc)) ) ).

% ATP.lambda_796
tff(fact_8338_ATP_Olambda__797,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_ez(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uuc)),aa(nat,nat,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_797
tff(fact_8339_ATP_Olambda__798,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_gs(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_798
tff(fact_8340_ATP_Olambda__799,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_eu(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_799
tff(fact_8341_ATP_Olambda__800,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,nat),Uua: fun(B,nat),Uub: A,Uuc: B] : aa(B,nat,aa(A,fun(B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_ado(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,Uu,Uub)),aa(B,nat,Uua,Uuc)) ).

% ATP.lambda_800
tff(fact_8342_ATP_Olambda__801,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_bd(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(C,B,Uua,Uuc)) ) ).

% ATP.lambda_801
tff(fact_8343_ATP_Olambda__802,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_xm(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_802
tff(fact_8344_ATP_Olambda__803,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_wy(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_803
tff(fact_8345_ATP_Olambda__804,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(C,A),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,product_prod(A,B),aa(C,fun(D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_ma(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),Uu),Uua),Uub),Uuc) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(D,B,Uua,Uuc)) ).

% ATP.lambda_804
tff(fact_8346_ATP_Olambda__805,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,D),Uub: A,Uuc: B] : aa(B,product_prod(C,D),aa(A,fun(B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_abq(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),Uu),Uua),Uub),Uuc) = aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,Uu,Uub)),aa(B,D,Uua,Uuc)) ).

% ATP.lambda_805
tff(fact_8347_ATP_Olambda__806,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_acg(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(A,bool,Uu,Uub))
        & pp(aa(B,bool,Uua,Uuc)) ) ) ).

% ATP.lambda_806
tff(fact_8348_ATP_Olambda__807,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_nn(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_807
tff(fact_8349_ATP_Olambda__808,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_ps(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),cos(real,aa(A,real,Uu,Uua))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_808
tff(fact_8350_ATP_Olambda__809,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: 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_pu(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),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,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uub)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% ATP.lambda_809
tff(fact_8351_ATP_Olambda__810,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: 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_qc(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),exp(real,aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_810
tff(fact_8352_ATP_Olambda__811,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: 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_qe(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),cos(real,aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_811
tff(fact_8353_ATP_Olambda__812,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_qt(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_812
tff(fact_8354_ATP_Olambda__813,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_rf(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),sqrt(aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% ATP.lambda_813
tff(fact_8355_ATP_Olambda__814,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: 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_qp(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),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,Uu,Uub)))) ) ).

% ATP.lambda_814
tff(fact_8356_ATP_Olambda__815,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pq(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),sqrt(aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ).

% ATP.lambda_815
tff(fact_8357_ATP_Olambda__816,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_un(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = divide_divide(real,real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc))),aa(A,B,Uu,Uub))),aa(A,B,Uua,Uuc))),real_V7770717601297561774m_norm(A,Uuc)) ) ).

% ATP.lambda_816
tff(fact_8358_ATP_Olambda__817,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(fun(A,C),fun(real,fun(A,bool)),aTP_Lamp_vx(fun(A,B),fun(fun(A,C),fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(A,C,Uua,Uuc))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uuc))),Uub))) ) ) ).

% ATP.lambda_817
tff(fact_8359_ATP_Olambda__818,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: filter(A),Uuc: A] : aa(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_uu(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(real,fun(B,B),real_V8093663219630862766scaleR(B),aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_ut(A,A)))))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_ut(A,A))))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_ut(A,A)))))) ) ).

% ATP.lambda_818
tff(fact_8360_ATP_Olambda__819,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_uq(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(real,fun(B,B),real_V8093663219630862766scaleR(B),aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))) ) ).

% ATP.lambda_819
tff(fact_8361_ATP_Olambda__820,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ur(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(real,fun(B,B),real_V8093663219630862766scaleR(B),aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))) ) ).

% ATP.lambda_820
tff(fact_8362_ATP_Olambda__821,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_lg(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_lf(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_821
tff(fact_8363_ATP_Olambda__822,axiom,
    ! [A: $tType,B: $tType,Uu: bool,Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_il(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(Uu)
        & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).

% ATP.lambda_822
tff(fact_8364_ATP_Olambda__823,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,set(B),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_adp(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = image2(C,B,Uua,aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),vimage(C,A,Uu,aa(set(A),set(A),insert(A,Uuc),bot_bot(set(A))))),Uub)) ).

% ATP.lambda_823
tff(fact_8365_ATP_Olambda__824,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(nat,fun(real,fun(A,bool)),aTP_Lamp_vd(fun(nat,A),fun(nat,fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fk(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uua))))) ) ) ).

% ATP.lambda_824
tff(fact_8366_ATP_Olambda__825,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,Uu: fun(C,D),Uua: fun(A,fun(B,C)),Uub: A,Uuc: B] : aa(B,D,aa(A,fun(B,D),aa(fun(A,fun(B,C)),fun(A,fun(B,D)),aTP_Lamp_hv(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D))),Uu),Uua),Uub),Uuc) = aa(C,D,Uu,aa(B,C,aa(A,fun(B,C),Uua,Uub),Uuc)) ).

% ATP.lambda_825
tff(fact_8367_ATP_Olambda__826,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(bool,fun(bool,A)),Uub: bool,Uuc: bool] : aa(bool,B,aa(bool,fun(bool,B),aa(fun(bool,fun(bool,A)),fun(bool,fun(bool,B)),aTP_Lamp_air(fun(A,B),fun(fun(bool,fun(bool,A)),fun(bool,fun(bool,B))),Uu),Uua),Uub),Uuc) = aa(A,B,Uu,aa(bool,A,aa(bool,fun(bool,A),Uua,Uub),Uuc)) ).

% ATP.lambda_826
tff(fact_8368_ATP_Olambda__827,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jd(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_827
tff(fact_8369_ATP_Olambda__828,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jb(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_828
tff(fact_8370_ATP_Olambda__829,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ajg(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ).

% ATP.lambda_829
tff(fact_8371_ATP_Olambda__830,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_dd(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_830
tff(fact_8372_ATP_Olambda__831,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_bn(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_831
tff(fact_8373_ATP_Olambda__832,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ajh(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ).

% ATP.lambda_832
tff(fact_8374_ATP_Olambda__833,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,B),Uub: D,Uuc: C] : aa(C,A,aa(D,fun(C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_mb(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(D,B,Uua,Uub)),Uuc)) ).

% ATP.lambda_833
tff(fact_8375_ATP_Olambda__834,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo7287701948861334536_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: fun(product_prod(B,B),bool),Uuc: A] :
          ( pp(aa(A,bool,aa(fun(product_prod(B,B),bool),fun(A,bool),aa(B,fun(fun(product_prod(B,B),bool),fun(A,bool)),aTP_Lamp_abr(fun(A,B),fun(B,fun(fun(product_prod(B,B),bool),fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(product_prod(B,B),bool,Uub,aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uu,Uuc)),Uua))) ) ) ).

% ATP.lambda_834
tff(fact_8376_ATP_Olambda__835,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,C),Uub: B,Uuc: D] : aa(D,A,aa(B,fun(D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_mc(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uub),aa(D,C,Uua,Uuc))) ).

% ATP.lambda_835
tff(fact_8377_ATP_Olambda__836,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),bool),aa(A,fun(B,fun(product_prod(A,B),bool)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_za(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_yz(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_836
tff(fact_8378_ATP_Olambda__837,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),bool),aa(A,fun(B,fun(product_prod(A,B),bool)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_yx(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_yw(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_837
tff(fact_8379_ATP_Olambda__838,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_wz(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Sup_Sup(A),image2(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_wy(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc),Uub)) ) ).

% ATP.lambda_838
tff(fact_8380_ATP_Olambda__839,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_xn(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Inf_Inf(A),image2(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_xm(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc),Uub)) ) ).

% ATP.lambda_839
tff(fact_8381_ATP_Olambda__840,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_ln(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_840
tff(fact_8382_ATP_Olambda__841,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_lt(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_841
tff(fact_8383_ATP_Olambda__842,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: C,Uub: fun(C,A),Uuc: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(C,fun(fun(C,A),fun(C,A)),aTP_Lamp_ql(fun(C,A),fun(C,fun(fun(C,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua))),aa(C,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(C,A,Uu,Uua)))) ) ).

% ATP.lambda_842
tff(fact_8384_ATP_Olambda__843,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_lp(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_843
tff(fact_8385_ATP_Olambda__844,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_lr(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).

% ATP.lambda_844
tff(fact_8386_ATP_Olambda__845,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_zm(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [Y5: C] :
          ( pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y5)),Uu))
          & pp(aa(set(product_prod(C,B)),bool,aa(product_prod(C,B),fun(set(product_prod(C,B)),bool),member(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y5),Uuc)),Uua)) ) ) ).

% ATP.lambda_845
tff(fact_8387_ATP_Olambda__846,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_yy(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),Uu),Uua),Uub),Uuc))
    <=> ? [A7: C] :
          ( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uua,A7)),aa(C,B,Uub,A7)) )
          & pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),A7),Uu)) ) ) ).

% ATP.lambda_846
tff(fact_8388_ATP_Olambda__847,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: fun(A,B),Uuc: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aa(fun(A,B),fun(product_prod(A,A),bool),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool)),aTP_Lamp_afj(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),Uu),Uua),Uub),Uuc))
    <=> ? [A17: A,A26: A] :
          ( ( Uuc = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A17),A26) )
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A17),Uu))
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A26),Uu))
          & pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uub,A17)),aa(A,B,Uub,A26))),Uua)) ) ) ).

% ATP.lambda_847
tff(fact_8389_ATP_Olambda__848,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),aTP_Lamp_ajc(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),aTP_Lamp_ajb(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uub),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uuc)) ).

% ATP.lambda_848
tff(fact_8390_ATP_Olambda__849,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,A),Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,A),fun(nat,fun(nat,A))),aTP_Lamp_aji(fun(A,fun(A,A)),fun(A,fun(fun(nat,A),fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),Uub),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uud),Uuc),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uud),Uuc)),Uuc))) ).

% ATP.lambda_849
tff(fact_8391_ATP_Olambda__850,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: A,Uub: A,Uuc: B,Uud: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_aaf(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = if(set(product_prod(C,B)),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uua),Uub),aa(set(product_prod(C,B)),set(product_prod(C,B)),insert(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_850
tff(fact_8392_ATP_Olambda__851,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: A,Uua: B,Uub: B,Uuc: C,Uud: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(C,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_zy(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = if(set(product_prod(A,C)),aa(B,bool,aa(B,fun(B,bool),fequal(B),Uua),Uub),aa(set(product_prod(A,C)),set(product_prod(A,C)),insert(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_851
tff(fact_8393_ATP_Olambda__852,axiom,
    ! [A: $tType,Uu: A,Uua: nat,Uub: fun(nat,A),Uuc: fun(nat,A),Uud: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aa(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),aTP_Lamp_aix(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uud),Uua),aa(nat,A,Uub,Uud),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uud),Uua),Uu,aa(nat,A,Uuc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uud),aa(nat,nat,suc,zero_zero(nat)))))) ).

% ATP.lambda_852
tff(fact_8394_ATP_Olambda__853,axiom,
    ! [A: $tType,B: $tType,I6: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I6),Uua: fun(I6,fun(A,B)),Uub: fun(I6,fun(A,B)),Uuc: A,Uud: A] : aa(A,B,aa(A,fun(A,B),aa(fun(I6,fun(A,B)),fun(A,fun(A,B)),aa(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B))),aTP_Lamp_qz(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(I6),B,aa(fun(I6,B),fun(set(I6),B),groups7311177749621191930dd_sum(I6,B),aa(A,fun(I6,B),aa(A,fun(A,fun(I6,B)),aa(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))),aa(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B)))),aTP_Lamp_qy(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))))),Uu),Uua),Uub),Uuc),Uud)),Uu) ) ).

% ATP.lambda_853
tff(fact_8395_ATP_Olambda__854,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_fi(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_fh(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_854
tff(fact_8396_ATP_Olambda__855,axiom,
    ! [Uu: nat,Uua: fun(nat,fun(real,real)),Uub: real,Uuc: nat,Uud: real] : aa(real,real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_pb(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu),Uua),Uub),Uuc),Uud) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Uua,Uuc),Uud)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_pa(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc)))),aa(real,real,aa(real,fun(real,real),times_times(real),Uub),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uud),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc)),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc)))))) ).

% ATP.lambda_855
tff(fact_8397_ATP_Olambda__856,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_fm(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fl(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud)) ) ).

% ATP.lambda_856
tff(fact_8398_ATP_Olambda__857,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ef(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uud)),Uua)),one_one(A))),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_857
tff(fact_8399_ATP_Olambda__858,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ee(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uuc),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_858
tff(fact_8400_ATP_Olambda__859,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_aat(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_trancl(A,Uub)))
          | ( Uuc = Uu ) )
        & ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),transitive_trancl(A,Uub)))
          | ( Uud = Uua ) ) ) ) ).

% ATP.lambda_859
tff(fact_8401_ATP_Olambda__860,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ft(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uub)),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_860
tff(fact_8402_ATP_Olambda__861,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_abl(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Uua),aa(set(A),set(A),insert(A,Uub),aa(set(A),set(A),insert(A,Uuc),aa(set(A),set(A),insert(A,Uud),bot_bot(set(A))))))),field2(A,Uu)))
        & ( ( ( Uua = Uuc )
            & ( Uub = Uud ) )
          | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A))))
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uuc)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))) )
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & ( Uua = Uuc )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))) ) ) ) ) ).

% ATP.lambda_861
tff(fact_8403_ATP_Olambda__862,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_aav(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_rtrancl(A,Uub)))
        & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),transitive_rtrancl(A,Uub))) ) ) ).

% ATP.lambda_862
tff(fact_8404_ATP_Olambda__863,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: A,Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_fh(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uud)),one_one(nat)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_863
tff(fact_8405_ATP_Olambda__864,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: 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_qr(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),Uu),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,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),one_one(nat)))) ) ).

% ATP.lambda_864
tff(fact_8406_ATP_Olambda__865,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_yb(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uuc),Uud))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uud),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)))
        & ! [I: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),divide_divide(nat,Uu,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))
           => ( ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I)),X_13))
            <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Uub),I)) ) )
        & ( ( Uuc = Uud )
         => ! [X4: vEBT_VEBT] :
              ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua)))
             => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_13)) ) )
        & ( ( Uuc != Uud )
         => ( vEBT_V5917875025757280293ildren(divide_divide(nat,Uu,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua,Uud)
            & ! [X4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)))
               => ( vEBT_V5917875025757280293ildren(divide_divide(nat,Uu,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua,X4)
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),X4))
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_865
tff(fact_8407_ATP_Olambda__866,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(product_prod(B,B),bool),Uuc: A,Uud: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(product_prod(B,B),bool),fun(A,fun(A,bool)),aa(fun(A,B),fun(fun(product_prod(B,B),bool),fun(A,fun(A,bool))),aTP_Lamp_abx(set(A),fun(fun(A,B),fun(fun(product_prod(B,B),bool),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uud),Uu))
             => pp(aa(product_prod(B,B),bool,Uub,aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uuc)),aa(A,B,Uua,Uud)))) ) ) ) ) ).

% ATP.lambda_866
tff(fact_8408_ATP_Olambda__867,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_ahy(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( ( Uub = Uuc )
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),Uu)) ) ) ).

% ATP.lambda_867
tff(fact_8409_ATP_Olambda__868,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),bool),Uua: A,Uub: A,Uuc: A,Uud: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_abv(fun(product_prod(A,A),bool),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> ( ( Uub = Uuc )
           => pp(aa(product_prod(A,A),bool,Uu,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud))) ) ) ) ).

% ATP.lambda_868
tff(fact_8410_ATP_Olambda__869,axiom,
    ! [C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(C,A),Uua: C,Uub: fun(C,A),Uuc: int,Uud: C] : aa(C,A,aa(int,fun(C,A),aa(fun(C,A),fun(int,fun(C,A)),aa(C,fun(fun(C,A),fun(int,fun(C,A))),aTP_Lamp_zr(fun(C,A),fun(C,fun(fun(C,A),fun(int,fun(C,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uub,Uud)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Uuc)),power_int(A,aa(C,A,Uu,Uua),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uuc),one_one(int))))) ) ).

% ATP.lambda_869
tff(fact_8411_ATP_Olambda__870,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(A),Uuc: set(C),Uud: B] :
          ( pp(aa(B,bool,aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_ads(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> ? [A7: A,B5: C] :
              ( ( Uud = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,A7)),aa(C,B,Uua,B5)) )
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A7),Uub))
              & pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B5),Uuc)) ) ) ) ).

% ATP.lambda_870
tff(fact_8412_ATP_Olambda__871,axiom,
    ! [A: $tType,B: $tType,I6: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: set(I6),Uua: fun(I6,fun(A,B)),Uub: fun(I6,fun(A,B)),Uuc: A,Uud: A,Uue: I6] : aa(I6,B,aa(A,fun(I6,B),aa(A,fun(A,fun(I6,B)),aa(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))),aa(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B)))),aTP_Lamp_qy(set(I6),fun(fun(I6,fun(A,B)),fun(fun(I6,fun(A,B)),fun(A,fun(A,fun(I6,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,aa(I6,fun(A,B),Uub,Uue),Uud)),aa(set(I6),B,aa(fun(I6,B),fun(set(I6),B),groups7121269368397514597t_prod(I6,B),aa(A,fun(I6,B),aTP_Lamp_qw(fun(I6,fun(A,B)),fun(A,fun(I6,B)),Uua),Uuc)),aa(set(I6),set(I6),aa(set(I6),fun(set(I6),set(I6)),minus_minus(set(I6)),Uu),aa(set(I6),set(I6),insert(I6,Uue),bot_bot(set(I6)))))) ) ).

% ATP.lambda_871
tff(fact_8413_ATP_Olambda__872,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: fun(C,A),Uue: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_qi(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uua,Uue)),aa(C,A,Uuc,Uub))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uu,Uub)),aa(C,A,Uud,Uue))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uuc,Uub)),aa(C,A,Uuc,Uub))) ) ).

% ATP.lambda_872
tff(fact_8414_ATP_Olambda__873,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_yw(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud)),Uu))
        | ( ( Uub = Uud )
          & pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue)),Uua)) ) ) ) ).

% ATP.lambda_873
tff(fact_8415_ATP_Olambda__874,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,real),Uud: fun(A,real),Uue: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_rb(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu,Uub),aa(A,real,Uuc,Uub))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uud,Uue)),ln_ln(real,aa(A,real,Uu,Uub)))),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uue)),aa(A,real,Uuc,Uub)),aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_874
tff(fact_8416_ATP_Olambda__875,axiom,
    ! [A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(D,A),Uua: fun(D,A),Uub: D,Uuc: fun(D,A),Uud: fun(D,A),Uue: D] : aa(D,A,aa(fun(D,A),fun(D,A),aa(fun(D,A),fun(fun(D,A),fun(D,A)),aa(D,fun(fun(D,A),fun(fun(D,A),fun(D,A))),aa(fun(D,A),fun(D,fun(fun(D,A),fun(fun(D,A),fun(D,A)))),aTP_Lamp_qa(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,A),fun(fun(D,A),fun(D,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uu,Uub)),aa(D,A,Uud,Uue))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uua,Uue)),aa(D,A,Uuc,Uub))) ) ).

% ATP.lambda_875
tff(fact_8417_ATP_Olambda__876,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: fun(C,A),Uue: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_qv(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(C,A,Uu,Uub))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(C,A,Uuc,Uub))),aa(C,A,Uud,Uue))),aa(A,A,inverse_inverse(A),aa(C,A,Uuc,Uub))))),divide_divide(A,aa(C,A,Uua,Uue),aa(C,A,Uuc,Uub))) ) ).

% ATP.lambda_876
tff(fact_8418_ATP_Olambda__877,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_yz(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
    <=> ( ( Uub = Uud )
        & pp(aa(A,bool,Uu,Uud))
        & pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue)),aa(A,set(product_prod(B,B)),Uua,Uud))) ) ) ).

% ATP.lambda_877
tff(fact_8419_ATP_Olambda__878,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT] : aa(vEBT_VEBT,B,aa(list(vEBT_VEBT),fun(vEBT_VEBT,B),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B))),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),aTP_Lamp_aiq(fun(A,B),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(A,B,Uu,aa(vEBT_VEBT,A,aa(list(vEBT_VEBT),fun(vEBT_VEBT,A),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))),Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_878
tff(fact_8420_ATP_Olambda__879,axiom,
    ! [B: $tType,A: $tType,Uu: bool,Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ne(bool,fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(Uu) ) ).

% ATP.lambda_879
tff(fact_8421_ATP_Olambda__880,axiom,
    ! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_acz(set(D),fun(C,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_880
tff(fact_8422_ATP_Olambda__881,axiom,
    ! [B: $tType,D: $tType,Uu: set(D),Uua: B] : aa(B,set(D),aTP_Lamp_adb(set(D),fun(B,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_881
tff(fact_8423_ATP_Olambda__882,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_adq(set(C),fun(B,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_882
tff(fact_8424_ATP_Olambda__883,axiom,
    ! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_adc(set(C),fun(A,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_883
tff(fact_8425_ATP_Olambda__884,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: C] : aa(C,set(B),aTP_Lamp_ajp(set(B),fun(C,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_884
tff(fact_8426_ATP_Olambda__885,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_aci(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_885
tff(fact_8427_ATP_Olambda__886,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_adg(set(A),fun(A,set(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_886
tff(fact_8428_ATP_Olambda__887,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_adi(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_887
tff(fact_8429_ATP_Olambda__888,axiom,
    ! [C: $tType,A: $tType,Uu: set(A),Uua: C] : aa(C,set(A),aTP_Lamp_ajo(set(A),fun(C,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_888
tff(fact_8430_ATP_Olambda__889,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_acm(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_889
tff(fact_8431_ATP_Olambda__890,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_acv(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_890
tff(fact_8432_ATP_Olambda__891,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( zero(Aa)
        & topological_t2_space(Aa)
        & topolo8386298272705272623_space(A) )
     => ! [Uu: Aa,Uua: A] : aa(A,Aa,aTP_Lamp_sp(Aa,fun(A,Aa),Uu),Uua) = Uu ) ).

% ATP.lambda_891
tff(fact_8433_ATP_Olambda__892,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_ais(C,fun(product_prod(A,B),C),Uu),Uua) = Uu ).

% ATP.lambda_892
tff(fact_8434_ATP_Olambda__893,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_px(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_893
tff(fact_8435_ATP_Olambda__894,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_mx(B,fun(C,B),Uu),Uua) = Uu ) ).

% ATP.lambda_894
tff(fact_8436_ATP_Olambda__895,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_xe(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_895
tff(fact_8437_ATP_Olambda__896,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: real] : aa(real,A,aTP_Lamp_aif(A,fun(real,A),Uu),Uua) = Uu ) ).

% ATP.lambda_896
tff(fact_8438_ATP_Olambda__897,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gm(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_897
tff(fact_8439_ATP_Olambda__898,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ph(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_898
tff(fact_8440_ATP_Olambda__899,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ax(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_899
tff(fact_8441_ATP_Olambda__900,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_mg(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_900
tff(fact_8442_ATP_Olambda__901,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_kc(A,fun(B,A)),Uu),Uua) = Uu ).

% ATP.lambda_901
tff(fact_8443_ATP_Olambda__902,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,B,aa(A,fun(B,B),aTP_Lamp_ka(A,fun(B,B)),Uu),Uua) = Uua ).

% ATP.lambda_902
tff(fact_8444_ATP_Olambda__903,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_im(A,fun(B,bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_903
tff(fact_8445_ATP_Olambda__904,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_bu(nat,nat),Uu) = Uu ).

% ATP.lambda_904
tff(fact_8446_ATP_Olambda__905,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_cm(int,int),Uu) = Uu ).

% ATP.lambda_905
tff(fact_8447_ATP_Olambda__906,axiom,
    ! [C: $tType] :
      ( topological_t2_space(C)
     => ! [Uu: C] : aa(C,C,aTP_Lamp_vb(C,C),Uu) = Uu ) ).

% ATP.lambda_906
tff(fact_8448_ATP_Olambda__907,axiom,
    ! [B: $tType,Uu: B] : aa(B,B,aTP_Lamp_abp(B,B),Uu) = Uu ).

% ATP.lambda_907
tff(fact_8449_ATP_Olambda__908,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ut(A,A),Uu) = Uu ) ).

% ATP.lambda_908
tff(fact_8450_ATP_Olambda__909,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_uw(A,A),Uu) = Uu ) ).

% ATP.lambda_909
tff(fact_8451_ATP_Olambda__910,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_va(A,A),Uu) = Uu ) ).

% ATP.lambda_910
tff(fact_8452_ATP_Olambda__911,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ni(A,A),Uu) = Uu ) ).

% ATP.lambda_911
tff(fact_8453_ATP_Olambda__912,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ae(A,A),Uu) = Uu ) ).

% ATP.lambda_912
tff(fact_8454_ATP_Olambda__913,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ahx(A,A),Uu) = Uu ) ).

% ATP.lambda_913
tff(fact_8455_ATP_Olambda__914,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_abo(A,A),Uu) = Uu ).

% ATP.lambda_914
tff(fact_8456_ATP_Olambda__915,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(bool),aTP_Lamp_ajv(A,set(bool)),Uu) = top_top(set(bool)) ).

% ATP.lambda_915
tff(fact_8457_ATP_Olambda__916,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_ack(A,set(B)),Uu) = top_top(set(B)) ).

% ATP.lambda_916
tff(fact_8458_ATP_Olambda__917,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_acl(A,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_917
tff(fact_8459_ATP_Olambda__918,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_bz(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_918
tff(fact_8460_ATP_Olambda__919,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_gj(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_919
tff(fact_8461_ATP_Olambda__920,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_au(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_920
tff(fact_8462_ATP_Olambda__921,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_mm(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_921
tff(fact_8463_ATP_Olambda__922,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_py(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_922
tff(fact_8464_ATP_Olambda__923,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_afv(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_923
tff(fact_8465_ATP_Olambda__924,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ad(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_924
tff(fact_8466_ATP_Olambda__925,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_afl(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_925
tff(fact_8467_ATP_Olambda__926,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_jh(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_926
tff(fact_8468_ATP_Olambda__927,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_ahh(B,option(A)),Uu) = none(A) ).

% ATP.lambda_927
tff(fact_8469_ATP_Olambda__928,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_aba(A,option(B)),Uu) = none(B) ).

% ATP.lambda_928
tff(fact_8470_ATP_Olambda__929,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_js(nat,bool),Uu))
    <=> $false ) ).

% ATP.lambda_929
tff(fact_8471_ATP_Olambda__930,axiom,
    ! [B: $tType,Uu: B] :
      ( pp(aa(B,bool,aTP_Lamp_ahr(B,bool),Uu))
    <=> $false ) ).

% ATP.lambda_930
tff(fact_8472_ATP_Olambda__931,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_ye(A,bool),Uu))
    <=> $false ) ).

% ATP.lambda_931
tff(fact_8473_ATP_Olambda__932,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_jt(nat,bool),Uu))
    <=> $true ) ).

% ATP.lambda_932
tff(fact_8474_ATP_Olambda__933,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_yf(A,bool),Uu))
    <=> $true ) ).

% ATP.lambda_933
tff(fact_8475_ATP_Olambda__934,axiom,
    ! [B: $tType,Uu: B] : aa(B,fun(nat,nat),aTP_Lamp_aes(B,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_934
tff(fact_8476_ATP_Olambda__935,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_aas(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_935

% Type constructors (861)
tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1,axiom,
    ! [A9: $tType,A18: $tType] :
      ( ( finite_finite(A9)
        & finite_finite(A18) )
     => finite_finite(product_prod(A9,A18)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_2,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => finite_finite(option(A9)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_3,axiom,
    ! [A9: $tType,A18: $tType] :
      ( ( finite_finite(A9)
        & finite_finite(A18) )
     => finite_finite(sum_sum(A9,A18)) ) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_4,axiom,
    finite_finite(char) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_5,axiom,
    finite_finite(bool) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_6,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => finite_finite(set(A9)) ) ).

tff(tcon_fun___Finite__Set_Ofinite_7,axiom,
    ! [A9: $tType,A18: $tType] :
      ( ( finite_finite(A9)
        & finite_finite(A18) )
     => finite_finite(fun(A9,A18)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
    ! [A9: $tType,A18: $tType] :
      ( comple592849572758109894attice(A18)
     => counta4013691401010221786attice(fun(A9,A18)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A9: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => condit1219197933456340205attice(fun(A9,A18)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A9: $tType,A18: $tType] :
      ( counta3822494911875563373attice(A18)
     => counta3822494911875563373attice(fun(A9,A18)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A9: $tType,A18: $tType] :
      ( comple592849572758109894attice(A18)
     => comple592849572758109894attice(fun(A9,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A9: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounde4967611905675639751up_bot(fun(A9,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A9: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounde4346867609351753570nf_top(fun(A9,A18)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A9: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => comple6319245703460814977attice(fun(A9,A18)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A9: $tType,A18: $tType] :
      ( boolea8198339166811842893lgebra(A18)
     => boolea8198339166811842893lgebra(fun(A9,A18)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A9: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => comple9053668089753744459l_ccpo(fun(A9,A18)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A9: $tType,A18: $tType] :
      ( semilattice_sup(A18)
     => semilattice_sup(fun(A9,A18)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A9: $tType,A18: $tType] :
      ( semilattice_inf(A18)
     => semilattice_inf(fun(A9,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A9: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounded_lattice(fun(A9,A18)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A9: $tType,A18: $tType] :
      ( order_top(A18)
     => order_top(fun(A9,A18)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A9: $tType,A18: $tType] :
      ( order_bot(A18)
     => order_bot(fun(A9,A18)) ) ).

tff(tcon_fun___Countable_Ocountable,axiom,
    ! [A9: $tType,A18: $tType] :
      ( ( finite_finite(A9)
        & countable(A18) )
     => countable(fun(A9,A18)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A9: $tType,A18: $tType] :
      ( preorder(A18)
     => preorder(fun(A9,A18)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A9: $tType,A18: $tType] :
      ( lattice(A18)
     => lattice(fun(A9,A18)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A9: $tType,A18: $tType] :
      ( order(A18)
     => order(fun(A9,A18)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A9: $tType,A18: $tType] :
      ( top(A18)
     => top(fun(A9,A18)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A9: $tType,A18: $tType] :
      ( ord(A18)
     => ord(fun(A9,A18)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A9: $tType,A18: $tType] :
      ( bot(A18)
     => bot(fun(A9,A18)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A9: $tType,A18: $tType] :
      ( uminus(A18)
     => uminus(fun(A9,A18)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_8,axiom,
    condit1219197933456340205attice(int) ).

tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations(int) ).

tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Rings_Onormalization__semidom__multiplicative,axiom,
    normal6328177297339901930cative(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___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___Bit__Operations_Oring__bit__operations,axiom,
    bit_ri3973907225187159222ations(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
    topolo2564578578187576103pology(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
    semiri2026040879449505780visors(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
    linord181362715937106298miring(int) ).

tff(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
    topolo4211221413907600880p_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
    semido2269285787275462019factor(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___Rings_Oordered__comm__semiring,axiom,
    ordere2520102378445227354miring(int) ).

tff(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
    normal8620421768224518004emidom(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_9,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_10,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___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___Countable_Ocountable_11,axiom,
    countable(int) ).

tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
    idom_abs_sgn(int) ).

tff(tcon_Int_Oint___Orderings_Opreorder_12,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_13,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_14,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_15,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_16,axiom,
    uminus(int) ).

tff(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___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_17,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_18,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_19,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_20,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_21,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_22,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_23,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_24,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_25,axiom,
    normal6328177297339901930cative(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_26,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_27,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_28,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_29,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_30,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_31,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_32,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_33,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_34,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_35,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_36,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_37,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_38,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_39,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_40,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_41,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_42,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_43,axiom,
    semido2269285787275462019factor(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_44,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_45,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_46,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_47,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_48,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_49,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_50,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_51,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_52,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_53,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_54,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_55,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_56,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom_57,axiom,
    normal8620421768224518004emidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_58,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_59,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_60,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_61,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_62,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_63,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_64,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_65,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_66,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_67,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_68,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_69,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_70,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_71,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_72,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_73,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_74,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_75,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_76,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_77,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_78,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_79,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_80,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_81,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_82,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_83,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Countable_Ocountable_84,axiom,
    countable(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_85,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_86,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_87,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_88,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_89,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_90,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_91,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_92,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_93,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_94,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_95,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_96,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_97,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_98,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_99,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_100,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_101,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Power_Opower_102,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_103,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_104,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_105,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_106,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_107,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_108,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_109,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_110,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Nat_Osize_111,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_112,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_113,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_114,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_115,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_116,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_117,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_118,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_119,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_120,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_121,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_122,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_123,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_124,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_125,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_126,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_127,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_128,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_129,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_130,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_131,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_132,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_133,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_134,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_135,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_136,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_137,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_138,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_139,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_140,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_141,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_142,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_143,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_144,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_145,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_146,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_147,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_148,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_149,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_150,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_151,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_152,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_153,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_154,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_155,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_156,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_157,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_158,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_159,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_160,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_161,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_162,axiom,
    ab_group_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
    field_char_0(rat) ).

tff(tcon_Rat_Orat___Countable_Ocountable_163,axiom,
    countable(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__neq__one_164,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_165,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_166,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_167,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_168,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_169,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_170,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_171,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_172,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_173,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_174,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_175,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_176,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_177,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_178,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_179,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_180,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_181,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_182,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_183,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_184,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_185,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_186,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_187,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_188,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_189,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_190,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_191,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_192,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_193,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_194,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_195,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_196,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_197,axiom,
    ! [A9: $tType] : counta4013691401010221786attice(set(A9)) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_198,axiom,
    ! [A9: $tType] : condit1219197933456340205attice(set(A9)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_199,axiom,
    ! [A9: $tType] : counta3822494911875563373attice(set(A9)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_200,axiom,
    ! [A9: $tType] : comple592849572758109894attice(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_201,axiom,
    ! [A9: $tType] : bounde4967611905675639751up_bot(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_202,axiom,
    ! [A9: $tType] : bounde4346867609351753570nf_top(set(A9)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_203,axiom,
    ! [A9: $tType] : comple6319245703460814977attice(set(A9)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_204,axiom,
    ! [A9: $tType] : boolea8198339166811842893lgebra(set(A9)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_205,axiom,
    ! [A9: $tType] : comple9053668089753744459l_ccpo(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_206,axiom,
    ! [A9: $tType] : semilattice_sup(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_207,axiom,
    ! [A9: $tType] : semilattice_inf(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_208,axiom,
    ! [A9: $tType] : bounded_lattice(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_209,axiom,
    ! [A9: $tType] : order_top(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_210,axiom,
    ! [A9: $tType] : order_bot(set(A9)) ).

tff(tcon_Set_Oset___Countable_Ocountable_211,axiom,
    ! [A9: $tType] :
      ( finite_finite(A9)
     => countable(set(A9)) ) ).

tff(tcon_Set_Oset___Orderings_Opreorder_212,axiom,
    ! [A9: $tType] : preorder(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Olattice_213,axiom,
    ! [A9: $tType] : lattice(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oorder_214,axiom,
    ! [A9: $tType] : order(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Otop_215,axiom,
    ! [A9: $tType] : top(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oord_216,axiom,
    ! [A9: $tType] : ord(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Obot_217,axiom,
    ! [A9: $tType] : bot(set(A9)) ).

tff(tcon_Set_Oset___Groups_Ouminus_218,axiom,
    ! [A9: $tType] : uminus(set(A9)) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_219,axiom,
    counta4013691401010221786attice(bool) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_220,axiom,
    condit1219197933456340205attice(bool) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_221,axiom,
    counta3822494911875563373attice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_222,axiom,
    comple592849572758109894attice(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_223,axiom,
    topolo4958980785337419405_space(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_224,axiom,
    topolo1944317154257567458pology(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_225,axiom,
    bounde4967611905675639751up_bot(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_226,axiom,
    bounde4346867609351753570nf_top(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_227,axiom,
    comple6319245703460814977attice(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_228,axiom,
    topolo2564578578187576103pology(bool) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_229,axiom,
    boolea8198339166811842893lgebra(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_230,axiom,
    topological_t2_space(bool) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_231,axiom,
    comple9053668089753744459l_ccpo(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_232,axiom,
    semilattice_sup(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_233,axiom,
    semilattice_inf(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_234,axiom,
    bounded_lattice(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_235,axiom,
    order_top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_236,axiom,
    order_bot(bool) ).

tff(tcon_HOL_Obool___Countable_Ocountable_237,axiom,
    countable(bool) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_238,axiom,
    preorder(bool) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_239,axiom,
    linorder(bool) ).

tff(tcon_HOL_Obool___Lattices_Olattice_240,axiom,
    lattice(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder_241,axiom,
    order(bool) ).

tff(tcon_HOL_Obool___Orderings_Otop_242,axiom,
    top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oord_243,axiom,
    ord(bool) ).

tff(tcon_HOL_Obool___Orderings_Obot_244,axiom,
    bot(bool) ).

tff(tcon_HOL_Obool___Groups_Ouminus_245,axiom,
    uminus(bool) ).

tff(tcon_List_Olist___Countable_Ocountable_246,axiom,
    ! [A9: $tType] :
      ( countable(A9)
     => countable(list(A9)) ) ).

tff(tcon_List_Olist___Nat_Osize_247,axiom,
    ! [A9: $tType] : size(list(A9)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_248,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_249,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_250,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_251,axiom,
    ordere1937475149494474687imp_le(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinear__continuum__topology,axiom,
    topolo8458572112393995274pology(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_252,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_253,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_254,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_255,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_256,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_257,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_258,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_259,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_260,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_261,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_262,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom,
    real_V768167426530841204y_dist(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_263,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_264,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_265,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_266,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_267,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_268,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_269,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_270,axiom,
    semiri3467727345109120633visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
    real_V6157519004096292374lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
    real_V7819770556892013058_space(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_271,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_272,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_273,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_274,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_275,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_276,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_277,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_278,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_279,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_280,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_281,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_282,axiom,
    topological_t2_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___Rings_Osemidom__divide_307,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_308,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_309,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_310,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_311,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_312,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_313,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_314,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_315,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_316,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_317,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_318,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_319,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_320,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_321,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_322,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_323,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_324,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_325,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_326,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_327,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_328,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_329,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_330,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_331,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_332,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_333,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_334,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_335,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_336,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_337,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_338,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_339,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_340,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_341,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_342,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_343,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_344,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_345,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_346,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Rings_Oring_347,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_348,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_349,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_350,axiom,
    dvd(real) ).

tff(tcon_String_Ochar___Countable_Ocountable_351,axiom,
    countable(char) ).

tff(tcon_String_Ochar___Nat_Osize_352,axiom,
    size(char) ).

tff(tcon_Sum__Type_Osum___Countable_Ocountable_353,axiom,
    ! [A9: $tType,A18: $tType] :
      ( ( countable(A9)
        & countable(A18) )
     => countable(sum_sum(A9,A18)) ) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_354,axiom,
    ! [A9: $tType,A18: $tType] : size(sum_sum(A9,A18)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_355,axiom,
    ! [A9: $tType] : condit1219197933456340205attice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_356,axiom,
    ! [A9: $tType] : counta3822494911875563373attice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_357,axiom,
    ! [A9: $tType] : bounde4967611905675639751up_bot(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_358,axiom,
    ! [A9: $tType] : bounde4346867609351753570nf_top(filter(A9)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_359,axiom,
    ! [A9: $tType] : comple6319245703460814977attice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_360,axiom,
    ! [A9: $tType] : comple9053668089753744459l_ccpo(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_361,axiom,
    ! [A9: $tType] : semilattice_sup(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_362,axiom,
    ! [A9: $tType] : semilattice_inf(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_363,axiom,
    ! [A9: $tType] : bounded_lattice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_364,axiom,
    ! [A9: $tType] : order_top(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_365,axiom,
    ! [A9: $tType] : order_bot(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_366,axiom,
    ! [A9: $tType] : preorder(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_367,axiom,
    ! [A9: $tType] : lattice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_368,axiom,
    ! [A9: $tType] : order(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_369,axiom,
    ! [A9: $tType] : top(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_370,axiom,
    ! [A9: $tType] : ord(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_371,axiom,
    ! [A9: $tType] : bot(filter(A9)) ).

tff(tcon_Option_Ooption___Countable_Ocountable_372,axiom,
    ! [A9: $tType] :
      ( countable(A9)
     => countable(option(A9)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_373,axiom,
    ! [A9: $tType] : size(option(A9)) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_374,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_375,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_376,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_377,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_378,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_379,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_380,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_381,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_382,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_383,axiom,
    real_V768167426530841204y_dist(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_Oreal__algebra_392,axiom,
    real_V6157519004096292374lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_393,axiom,
    real_V7819770556892013058_space(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___Topological__Spaces_Ot2__space_401,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_402,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_403,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_404,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_405,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_406,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_407,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_408,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_409,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_410,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_411,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_412,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_413,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_414,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_415,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_416,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_417,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_418,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_419,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_420,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_421,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_422,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_423,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_424,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_425,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_426,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_427,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_428,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_429,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_430,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_431,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_432,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_433,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_434,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_435,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_436,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_437,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_438,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_439,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_440,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_441,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_442,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_443,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_444,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_445,axiom,
    dvd(complex) ).

tff(tcon_Typerep_Otyperep___Countable_Ocountable_446,axiom,
    countable(typerep) ).

tff(tcon_Typerep_Otyperep___Nat_Osize_447,axiom,
    size(typerep) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_448,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_449,axiom,
    counta4013691401010221786attice(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___Lattices_Obounded__semilattice__sup__bot_456,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_457,axiom,
    bounde4346867609351753570nf_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_458,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_459,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_460,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_461,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_462,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_463,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_464,axiom,
    comple9053668089753744459l_ccpo(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_465,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_466,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_467,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_468,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_469,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_470,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_471,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_472,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_473,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_474,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_475,axiom,
    semigroup_mult(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___Countable_Ocountable_484,axiom,
    countable(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_485,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_486,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_487,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_488,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_489,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_490,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_491,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_492,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_493,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_494,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_495,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Otop_496,axiom,
    top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_497,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Obot_498,axiom,
    bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_499,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_500,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_501,axiom,
    zero(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,
    ! [A9: $tType,A18: $tType] :
      ( ( topolo4958980785337419405_space(A9)
        & topolo4958980785337419405_space(A18) )
     => topolo4958980785337419405_space(product_prod(A9,A18)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_505,axiom,
    ! [A9: $tType,A18: $tType] :
      ( ( topological_t2_space(A9)
        & topological_t2_space(A18) )
     => topological_t2_space(product_prod(A9,A18)) ) ).

tff(tcon_Product__Type_Oprod___Countable_Ocountable_506,axiom,
    ! [A9: $tType,A18: $tType] :
      ( ( countable(A9)
        & countable(A18) )
     => countable(product_prod(A9,A18)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_507,axiom,
    ! [A9: $tType,A18: $tType] : size(product_prod(A9,A18)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_508,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_509,axiom,
    counta4013691401010221786attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_510,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_511,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_512,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_513,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_514,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_515,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_516,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_517,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_518,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_519,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_520,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_521,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_522,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_523,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_524,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable_Ocountable_525,axiom,
    countable(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_526,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_527,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_528,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_529,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_530,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_531,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_532,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_533,axiom,
    uminus(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_534,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_535,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_536,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_537,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_538,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_539,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_540,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_541,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_542,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_543,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_544,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_545,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_546,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_547,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_548,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_549,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_550,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_551,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_552,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_553,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_554,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_555,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_556,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_557,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_558,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_559,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_560,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_561,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_562,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_563,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_564,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_565,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_566,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_567,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_568,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_569,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_570,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_571,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_572,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_573,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_574,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_575,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_576,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_577,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_578,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_579,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_580,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_581,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_582,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_583,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_584,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_585,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_586,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_587,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_588,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_589,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_590,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_591,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_592,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_593,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_594,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_595,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_596,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_597,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_598,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_599,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_600,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_601,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_602,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_603,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_604,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_605,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_606,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_607,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_608,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_609,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_610,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_611,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_612,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_613,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_614,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_615,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_616,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_617,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_618,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_619,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_620,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_621,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_622,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_623,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_624,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_625,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_626,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_627,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_628,axiom,
    bit_un5681908812861735899ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_629,axiom,
    euclid5411537665997757685th_nat(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_630,axiom,
    ordere1937475149494474687imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_631,axiom,
    euclid3128863361964157862miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_632,axiom,
    euclid4440199948858584721cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_633,axiom,
    semiri6575147826004484403cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_634,axiom,
    strict9044650504122735259up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_635,axiom,
    ordere580206878836729694up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_636,axiom,
    ordere2412721322843649153imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_637,axiom,
    bit_se359711467146920520ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_638,axiom,
    linord2810124833399127020strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_639,axiom,
    strict7427464778891057005id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_640,axiom,
    ordere8940638589300402666id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_641,axiom,
    euclid3725896446679973847miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_642,axiom,
    semiri2026040879449505780visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_643,axiom,
    linord181362715937106298miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_644,axiom,
    linord8928482502909563296strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_645,axiom,
    semiri3467727345109120633visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_646,axiom,
    ordere6658533253407199908up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_647,axiom,
    ordere6911136660526730532id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_648,axiom,
    cancel2418104881723323429up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_649,axiom,
    cancel1802427076303600483id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_650,axiom,
    comm_s4317794764714335236cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_651,axiom,
    bit_semiring_bits(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_652,axiom,
    ordere2520102378445227354miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_653,axiom,
    cancel_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_654,axiom,
    linordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_655,axiom,
    ordered_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_656,axiom,
    linordered_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_657,axiom,
    ab_semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_658,axiom,
    semiring_1_cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_659,axiom,
    algebraic_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_660,axiom,
    comm_monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_661,axiom,
    comm_monoid_diff(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_662,axiom,
    ab_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_663,axiom,
    ordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_664,axiom,
    semiring_parity(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_665,axiom,
    comm_monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_666,axiom,
    semiring_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_667,axiom,
    comm_semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_668,axiom,
    comm_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_669,axiom,
    semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_670,axiom,
    semidom_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_671,axiom,
    semidom_divide(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_672,axiom,
    semiring_numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_673,axiom,
    semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_674,axiom,
    zero_less_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_675,axiom,
    comm_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_676,axiom,
    semiring_char_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_677,axiom,
    zero_neq_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_678,axiom,
    preorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_679,axiom,
    linorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_680,axiom,
    monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_681,axiom,
    monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_682,axiom,
    semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_683,axiom,
    semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_684,axiom,
    mult_zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_685,axiom,
    order(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_686,axiom,
    semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom_687,axiom,
    semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oord_688,axiom,
    ord(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Power_Opower_689,axiom,
    power(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Onumeral_690,axiom,
    numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ozero_691,axiom,
    zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oone_692,axiom,
    one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Odvd_693,axiom,
    dvd(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osize_694,axiom,
    size(code_natural) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_695,axiom,
    size(vEBT_VEBT) ).

tff(tcon_Record_Otuple__isomorphism___Nat_Osize_696,axiom,
    ! [A9: $tType,A18: $tType,A19: $tType] : size(tuple_isomorphism(A9,A18,A19)) ).

% Helper facts (24)
tff(help_If_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : if(A,fFalse,X,Y) = Y ).

tff(help_If_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : if(A,fTrue,X,Y) = X ).

tff(help_fEx_1_1_U,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A] :
      ( ~ pp(aa(A,bool,P2,X))
      | pp(aa(fun(A,bool),bool,fEx(A),P2)) ) ).

tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,P2: fun(A,bool),X: A] :
      ( ~ pp(fAll(A,P2))
      | pp(aa(A,bool,P2,X)) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P2: bool] :
      ( pp(P2)
      | pp(aa(bool,bool,fNot,P2)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P2: bool] :
      ( ~ pp(aa(bool,bool,fNot,P2))
      | ~ pp(P2) ) ).

tff(help_COMBB_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(B,C),Q: fun(A,B),R2: A] : aa(A,C,combb(B,C,A,P2,Q),R2) = aa(B,C,P2,aa(A,B,Q,R2)) ).

tff(help_COMBC_1_1_U,axiom,
    ! [A: $tType,C: $tType,B: $tType,P2: fun(A,fun(B,C)),Q: B,R2: A] : aa(A,C,combc(A,B,C,P2,Q),R2) = aa(B,C,aa(A,fun(B,C),P2,R2),Q) ).

tff(help_COMBS_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(A,fun(B,C)),Q: fun(A,B),R2: A] : aa(A,C,combs(A,B,C,P2,Q),R2) = aa(B,C,aa(A,fun(B,C),P2,R2),aa(A,B,Q,R2)) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(fconj(P2,Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(fconj(P2,Q))
      | pp(P2) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(P2)
      | ~ pp(Q)
      | pp(fconj(P2,Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(fdisj(P2,Q))
      | pp(P2)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P2: bool] :
      ( ~ pp(Q)
      | pp(fdisj(P2,Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(P2)
      | pp(fdisj(P2,Q)) ) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P2: bool] :
      ( ( P2 = fTrue )
      | ( P2 = fFalse ) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ( X != Y )
      | pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P2: fun(A,bool)] : aa(A,bool,P2,fChoice(A,P2)) = aa(fun(A,bool),bool,fEx(A),P2) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P2),Q))
      | ~ pp(P2)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P2: bool] :
      ( ~ pp(Q)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P2),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P2: bool,Q: bool] :
      ( pp(P2)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P2),Q)) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    info = aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)) ).

%------------------------------------------------------------------------------